Laser Chem. 1983, Vol. 4, pp. 1-104
0278--6273/83/0402-0001512.00/0
(C) harwood academic publishers gmbh
Printed in Great Britain
Multiple-Photon Infrared Laser
Photophysics and
Photochemistry, II
V. N. BAGRATASHVILI, V. S. LETOKHOV, A. A. MAKAROV and E. A.
RYABOV
Institute of Spectroscopy, USSR Academy of Sciences, 142092, Troitsk,
Moscow Region, USSR
(Received June 15, 1982)
This is the second part of a review paper on the multiple-photon excitation of molecules by
intense infrared laser light. This part commences with Section 3.
Contents
3. Elements of Theory of Resonant Interactions of IR
Monochromatic Fields with Molecules
3.1. Introduction to theoretical treatment
3.1.1. Schr6dinger equation and approximations
3.1.2. Quasienergetic or dressed quantum states
3.2. Coherent excitation of various quantum systems
3.2.1. Two-level system
3.2.2. Harmonic oscillator
3.2.3. Multiphoton transitions for multilevel system
3.2.4. Anharmonic oscillator
3.3. Incoherent excitation of quantum system with relaxation
3.3.1. Conditions of incoherent excitation for two-level
system
3.3.2. Effect of rotational bottleneck
3
3
3
5
6
6
13
17
25
28
29
34
V. N. BAGRATASHVILI ET AL.
3.4. Excitation from a discrete level into a quasicontinuous
band
3.4.1. Derivation of the kinetic equations
3.4.2. Validity of the kinetic equations
3.5. Excitation of the vibrational quasicontinuum
3.5.1. Kinetic equations for stimulated transitions in
vibrational quasicontinuum
3.5.2. Excitation for the case of constant cross-sections
3.5.3. Excitation for the case of varying cross-sections
3.5.4. Influence of dissociative decay and rotational
structure
3.5.5. Multiphoton transitions into quasicontinuum
3.6. Concluding remarks
4. Physics of Interaction of a Polyatomic Molecule with an
Intense IR Field
4.1. Main stages of MP excitation of polyatomic molecules
by IR radiation
4.1.1. Role of IR radiation intensity in excitation of
lower vibrational levels
of IR radiation energy fluence in excitation
Role
4.1.2.
of vibrational quasicontinuum
4.2. MP excitation of molecules at lower vibrational
transitions
4.2.1. "Nonsaturable" behavior of MP absorption of IR
radiation by polyatomic molecules
4.2.2. Resonant character of MP absorption
4.2.3. Fraction of MP excited molecules
4.3. MP excitation of molecules in the vibrational
quasicontinuum
4.3.1. Evolution of MP absorption spectrum in the
vibrational quasicontinuum
4.3.2. Distribution of vibrational energy of a molecule
under MP excitation
4.4. Dissociation and MP excitation of molecules in
continuum
4.4.1. "Threshold" character of dissociation yield
4.4.2. Overexcitation of polyatomic molecules in
continuum
4.5. MP excitation of the electronic states of polyatomic
molecules by intense IR radiation
39
40
45
46
47
51
55
58
59
62
63
63
64
66
69
70
73
76
80
80
85
90
90
95
98
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
3. ELEMENTS OF THEORY OF RESONANT INTERACTIONS
OF IR MONOCHROMATIC FIELDS WITH MOLECULES
3.1. Introduction to theoretical treatment
The action of IR laser radiation on gases of polyatomic molecules comprises
a substantial number of the fundamental questions dealing with the interaction of an isolated molecule with radiation calling for adequate theoretical
treatment.
3.1.1. Schr6dinger equation and approximations
The classical treatment of laser-radiation fields seems to be a common
starting point for discussions with the following form of the operator of
molecule-field interaction:
(z(t)
l),E(t)
(3.1)
quite substantiated theoretically (see, for example, Ref. 174) where I), is
the dipole-moment operator of a molecule. The dependence of the electric
component E of the field on spatial coordinates can be neglected, since
the radiation wavelength is many orders larger than the molecular size.
The validity of this description of laser radiation in terms of a monochromatic field is discussed in Section 2.8 from our previous paper, 175 but as
far as the theory is concerned, we shall follow in this series of papers just
this approximation, since it makes the mathematics much easier. Of course,
exceptions arising from radiation nonmonochromaticity will be qualitatively discussed when necessary. Thus, neglecting the spatial dependence
8’ cos
of E, let us take the electric field acting on the molecule as E
f where the amplitude 8’ is constant or, at least, slowly varying in time.
Then the wavefunction of the molecule evolves in time in accordance with
the Schr6dinger equation
ih
10_._:_
0t
/?/o
(I), 8’cosft),
(3.2)
where -f-/o is the Hamiltonian of the molecule without an external field.
The description in terms of the wavefunction is absolutely equivalent to
that in terms of the probability amplitudes referring to different quantum
states of the stationary Hamiltonian Ho of the molecule. Indeed, if one
presents the wavefunction (t) as the expansion into series by stationary
eigenfunctions Oj or/-/o, i.e.,
V. N. BAGRATASHVILI ET AL.
(t)
aj(t)qJexp
t it
-Ejt
(3.3)
then (t) can be fully determined through the temporal behavior of the
probability amplitudes a(t). It is easy to derive the equation for the amplitude ak(t) of the level ]k). Substitute the expansion (3.3) into Eq. (3.2),
multiply both sides by
Okexp(-Ekt), integrate over the internal molecular
coordinates, and use the orthogonality of stationary wavefunctions. Then
one comes to
dok
dt
i lka{exp[i(l)
tOok)t] + exp[--i(O + tOOk)t]},
(3.4)
jk
where /k
(kll[j)/2h. There are no terms with j k in sum, since all
directions are, in particular, equivalent in nondegenerate states. Therefore,
diagonal matrix elements of the dipole-moment operator vanish, and, for
degenerate states, one can always choose the basis displaying the same
property. Note also that the equality "k *k follows from the Hermiticity
of the operator I, and, since one may choose wavefunctions of stationary
states to be real, then the equality k;
"jk can be taken without any
additional reservations.
It can be seen from Eq. (3.4) that, even if their solutions may possibly
be found, one can know in any case the frequencies and matrix elements
of I for transitions between molecular levels. It is clear from the discussion
of the vibration-rotation spectra of polyatomic molecules in Section 2 of
our previous paper 75 how hard the problems arising here can be. Nevertheless, just as theory gradually reveals the principal rules and features of
the structure of molecular spectra starting from simple to more complex,
the main features of appropriate spectroscopic models of excitation by
monochromatic fields may be traced starting similarly from the simple.
One should keep in mind one more simplification allowed in these processes
due to the resonant character of excitation processes. In particular, it is
possible as a good approximation to take into consideration not all transitions but rather only those which are close to resonance with the field.
In this Section we shall work with Eq. (3.4) only for the simplest spectroscopic models. Some more complex models appropriate to more detailed
treatments of vibration-rotation spectra will be discussed in the subsequent
Sections, in close connection with experimental results.
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
The topic of the interaction of radiation with an isolated molecule embraces nearly all the problems important for our goals. However, to understand some points, it is essential to take into account relaxation processes, and the simplest ideas will be discussed in Section 3.4. Other points
related to the excitation of a gas as a whole (rather than an individual
molecule) seem to be less essential. For our goals, the statistical distribution
of molecules in the gas becomes only the averaging over initial conditions.
But it should be noted that, in view of coherent transient effects (see, for
example, Ref. 176) or nonlinear effects in gas media (see, for example,
Ref. 177), the inclusion of a velocity distribution responsible for Doppler
broadening becomes of fundamental importance. Also, we are not going
to consider the influence of the medium on the radiation which can result
in various propagation effects (see, for example, Refs. 178-180); i.e., a
medium is believed to be optically thin in our theoretical treatment.
Finally, we will not take into account the spontaneous decay of excited
molecular levels, since the typical radiative lifetime of about 10-1-10-3 s
in the IR range is several orders longer than the laser pulses used in
experiments on IR excitation of molecules. We must stress, however, that
if the lifetime becomes comparable with the pulse duration new qualitative
effects will appear (see, for example, Ref. 181).
3.1.2. Quasienergetic or dressed quantum states
Before considering specific problems we wish to note rather a common
feature of solutions of the nonstationary Schr6dinger equation (3.2). This
feature is that (see, for example, Ref. 182), if 8’ const, eigenfunctions
of Eq. (3.2) may be presented in the following form:
(t,q)
qdt, q)exp(-iet),
(3.5)
where q are periodic functions of time, their cycle being just equal to
2r/I. By their form, the eigenfunctions (3.5) resemble the conventional
representation of stationary wavefunctions, and so the eigenvalues e are
called quasienergies, and eigenstates (3.5) are called quasienergetic or,
more commonly, dressed states. Since multiplying the function q by the
factor exp(intt), where n is an arbitrary integer, does not affect the property
of periodicity, then one may conclude that quasienergies are defined with
the accuracy to an arbitrary term nO. The quasienergy approach is very
useful, since some features of dressed states are very similar to those of
ordinary stationary levels. In particular, resonances of spontaneous emis-
V. N. BAGRATASHVILI ET AL.
sion as well as resonances of absorption and Raman scattering in second
weak probe field are attached now to quasienergies. 8Another important fact (see Ref. 182) can be discovered for the case
when the amplitude of external field changes very slowly (adiabatically)
from zero to some finite value. If initially (before the field is switched on)
the system is in a definite stationary state li), then at every subsequent
instant of time the system is in that dressed state which adiabatically
originates from the primary one, i.e., comes to li) when --* 0. Of course,
during this process the quasienergy itself as well as the dressed-state wavefunction will depend parametrically on the current amplitude of external
field.
If, on the contrary, we are interested in another situation when the field
0, then, in the general
amplitude is a step function being switched on at
case, the dependence of the wavefunction on time is described by the
superposition of dressed-state eigenfunctions:
’
’
(t)
(3.6)
ckq0kexp(- iekt),
k
where the factors c must satisfy the initial condition at
0.
3.2. Coherent excitation of various quantum systems
Let us start from the consideration of various simple quantum systems
which are simplified models of a molecular system.
3.2.1. Two-level system
The first natural step is the consideration of a two-level system to obtain
the simplest estimates for the excitation efficiency of the quantum system.
In this case, the general equation (3.4) which describes the dynamics of
probability amplitudes ao and a of the lower and upper levels, correspondingly, take the following form:
dao
dal
i’yoa{exp[i(
tOo)t]
+ exp[-i(O + tOo)t]}
(3.7)
ilolao{exp[-i(
tolo)t]
-b
exp [i(1 + tO]ot]}
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
We are interested, of course, in the situation when the radiation frequency
II is close to that Oo of transition, i.e.,
Ill
at least the condition
O1o1 ’ l-I
(3.8)
is met. If so, then one can easily see that the exponential terms in the
fight-hand sides of Eqs. (3.7) differ greatly in frequency. To estimate
relative contributions from these terms, let us first assume that perturbation
and a(t 0) 0, at every
theory can be used, i.e., with ao(t 0)
1. Accepting ao
subsequent instant of time la l
in the second of
Eqs. (3.7) and integrating, we get
-
/o
al
OJlO
+
yo_.
{exp[
i(1
Oo)t]
{exp[i(12 + tOo)tl
1}
(3.9)
1}
It can easily be seen that, if condition (3.8) is fulfilled, the second term
is much smaller than the first one. Therefore, one has to expect that the
effect of an excitation of a two-level system arises mainly from the exponential term with the difference frequency.
The approximation in which the exponential terms with frequency 12 +
Oo are removed from Eqs. (3.7) is usually referred to as the rotatingwave approximation.* Besides the requirement (3.8), the second criterion
for using the rotating-wave approximation is the condition
Io’l ,
htOlo
(3.10)
For the transitions in the IR range, this condition is valid up to very strong
radiation fields, the intensities of which are many orders higher than those
ordinarily used in experiments. For example, taking the transition frequency 1000 crn and the typical value IXo 0.3 Debye of dipole moment
for some allowed vibration-rotation transition, one can find that the equality
-
*This name came from the visual vector model for the process of excitation of a two-level
system introduced into theory in Ref. 183 by analogy with the nuclear magnetic resonance.
V. N. BAGRATASHVILI ET AL.
c2/8,1T
requires an enormous intensity of radiation I
2" 10 TM W/cm2.
We shall adhere below to the rotating-wave approximation without any
additional reservation. In this approximation, Eqs. (3.7) take the form
l,Ol/holo
dao
i/olalexp [i(
dt
olo)t)]
(3.11)
dal
i/olaoexp [- i(f
tolo)t]
the analytical solution of which is quite simple. Two independent solutions
of Eqs. (3.1 l) may be presented, for example, as
1/2
1)
_(1-213 1) sign( /o-1O)10)
q-
2
12
exp
e --i(n
tOlO)t
[
tolO
2
(13-
1)t]
(3.12)
sign
(.12 -.tolO
"tOl
exp
e-i(fl-tl)t
2
Oo( 13
+ 1)t
/
(3.13)
where 13
(O10)2] 1/2. Through direct substitution, one
[1 + 4/o21/(1
can make sure that these expressions really give solutions of Eqs. (3.11).
Of course, the written-out solutions (3.12) and (3.13) comply with the
1. By taking the supernormalization requirement lao(t)l 2 + lal(t)l 2
positions of (3.12) and (3.13) one can combine any other solutions. But
it is just each of solutions (3.12) and (3.13) that is eigen in the sense that
it corresponds to a definite value of quasienergy.
]
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
9
Indeed, the time-dependent wavefunction of two-level system has the
form
-qt(t)
(2)
exp
-i
for the solution (3.12), and
1/9-
sign
+o
213
(’ tle-itt,]
+
/ol
(3.14)
/
(13- 1)
2
I(13-1213 ) 1/21j0
le-iltql
213
exp{-i[Eh "-21( + 1)It}
(t)
sign
Z
(3.15)
/ol
for the solution (3.13). The comparison with the general form (3.5) of
dressed states convinces us that the first of our solutions (3.12) corresponds
to the quasienergy
eo
a
oolO
2
{[
l
+ (D, 4/’
O)lO)2J
’/
}
(3.16)
and the second one to the quasienergy
e
D,
COco
2
{[
if they are counted off the energy
+ (12
l
0010) 2j
4/’
’/e
+
Eo of the lower level.
}
(3.17)
10
V. N. BAGRATASHVILI ET AL.
FIGURE 3.1 The diagram of dressed states when a near-resonant field is applied to a twolevel system. (a) Shift of levels due to the dynamic Stark effect. (b) Splitting of levels
by exactly resonant field.
The diagram of dressed states is shown in Figure 3.1. If the field frequency is not equal exactly to the transition frequency, and the field am1, the dressed state (3.14) comes to the
plitude tends to zero, then 13
lower state, and the dressed state (3.15) comes to the upper state. The
shift of levels in the presence of field is usually referred to as the shift due
to the dynamic Stark effect. The case when the field frequency is exactly
equal to the transition frequency is called degenerate. In this case, the
quasienergies are just equal to eo,
+--/o; i.e., each of the dressed states
cannot be attributed to any of two stationary levels. Therefore, one may
often hear that the exactly resonant field leads to a splitting of levels.
Now let us trace the dynamics of excitation of a two-level system assuming that initially the system is in the lower state, and the field intensity
is the step function being switched on at t
0. To do it one should [see
the discussion on Eq. (3.6)] make up a superposition of solutions (3.12)
and (3.13) which gives ao
0. As a result, we get
and a
0 at
the following dependencies of probability amplitudes on time:
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
11
and, for the probabilities of finding the system at the lower and upper
levels (or, in other words, for the relative populations of levels), we have
a(t)ao(t)=
Poo(t)
tolo) 2
a*l(t)a(t)
9a(t)
+ 4/o21
(.
11)10)2+
f
0o) 2
(f/
(-
cos2{[(
sin 2 [(fl
4/gl
0110)2 +
4/o2]/2t)
tolO) 2
4021
sin2
+ 4/o21]
{[(1"
1/2t}
0)10) 2 +
41111/2t}
It may be seen from Eqs. (3.19) that the populations oscillate about their
averaged values
poo(t)
Pl l(t)
1-
(",
(’
(010) 2
0)10) 2 "1"
4/021
(3.20)
+ 4/o2
In the case of exact resonance, i.e., with ll
tolO, and averaged values
of relative populations of both levels are just equal to 1/2, and the frequency
of oscillation of probability amplitudes, the so-called Rabi frequency, is
"YOl With the detuning from exact resonance, it may be seen that the
condition
I.
21",/Ol > a
tolOl
(3.21)
12
V. N. BAGRATASHVILI ET AL.
serves as the criterion of effective excitation, or, in other words, the halfwidth of the resonance at the half-maximum is equal to 2 /ol Therefore,
the value 2 /ol is also called the power broadening.
Let us turn now to estimates. Take, for example, the intensity of the
radiation field to be 107 W/cma, and the transition dipole moment to be
101 s-1. This means
0.3 Debye. Then the Rabi frequency /ol is 4.1
that in the case of exact resonance the time interval of the first semicycle
of oscillations required for full inversion of populations is just "rr/2 /o
3.8
10-11 S. At the same time, with moving away exact resonance,
the characteristic detuning f
2 /Ol (which still allows us
O)1o
to treat the excitation of upper level as rather effective) is 0.44 cm-1 Now
it is clear (see the discussion in Section 2.2.1 from our previous paper 175)
why, in view of the ordinary absorption spectroscopy which examines the
resonances, one should use weak fields. Really, in the general case, even
though the source is monochromatic, the spectral resolution drops as its
intensity increases.
Thus, we have considered the dynamics of excitation of a two-level
system at instantaneous switching on of the radiation field with the constant
amplitude. In case of the exact resonance, the obtained expressions remain
valid for fields with time-dependent amplitude, but only with the substitution.
I.
’t
ff(’r)da"
(3.22)
In the general case of detuning from exact resonance, it is impossible to
obtain analytical solutions. However, in the limit of adiabatically slow
switching on of the field, according to the general principles [see the
discussion following Eq. (3.5)], the state of the system being initially at
the lower level will be described at all further instants by the solution
(3.12) which corresponds to the quasienergy (3.16), the current value of
field amplitude being put on. The criterion of adiabaticity here is the
condition
(3.23)
where a’sw is the characteristic switching on time of the field. The criterion
of excitation efficiency (3.21) remains valid for this case though_the resonance half-width is somewhat smaller and comes to 2 /ol I/X/3.
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
13
Let us turn now to a discussion of the possible effects of nonmonochromaticity of radiation. It is hard, of course, to speak here about population
oscillations, since they arise from the coherent nature of excitation. As far
as the efficiency of upper-level excitation is concerned, in the case of
detuning from the exact resonance which exceeds the width of the radiation
spectrum, the estimates obtained remain valid, but one should use the
modified Rabi frequency /god
(iXo/h)(27rl/c)l/2, where I means the total
field intensity integrated over its spectrum. When the field spectrum overlaps the transition frequency, the resonant part of the spectrum may be
mainly responsible for excitation. Therefore, to estimate the time required
for excitation one should take the intensity of that part of the field spectrum
which lies within the value of power broadening. Thus, if the power
broadening exceeds the width of the field spectrum, then one may use the
above formulas to estimate the excitation time. Actually this calls for rather
moderate intensities of radiation. For example, with a transition dipole
moment of 0.3 Debye and spectrum width of (Al))rad
0.03 crn-1, the
equality 2 /Ol
(Al)rad is reached at the intensity 4.6 x 104 W/cm2.
One more point should be noted. If the modes of radiation of laser are
synchronized (see Section 2.8 from Ref. 175), the intensity at some instant
may essentially exceed its mean value. This should be taken into account
in estimates.
3.2.2. Harmonic oscillator
The harmonic oscillator is the simplest approximation for the levels of an
individual vibrational mode of a molecule. In view of behaviors in external
nonstationary fields, the harmonic oscillator is also the simplest case, since
it is possible to obtain analytical solutions of the Schr6dinger equation for
an arbitrary dependence of the field amplitude on time (see, for example,
Ref. 184). We shall consider here only the solution for the case of a
monochromatic field and, for simplicity, adhere to the rotating-wave approximation. The exact solution, which is much more cumbersome, gives
only small corrections. So, let us consider Eq. (3.4) for the probability
amplitudes of an infinite set of levels of the harmonic oscillator. Since,
within the harmonic approximation, only transitions between adjacent levels are allowed, we get the set of equations
dav
d---’
il{v/2av
exp[- i(1
to)t]
+ (v + 1)/2a,
+
exp[i(
to)t]
(3.24)
14
V. N. BAGRATASHVILI ET AL.
where /
/ol, the dependence of the dipole-moment matrix element on
the number of transition is taken into account [see Eq. (2.44) from Ref.
175], and only the terms with the difference frequency are retained. Since
all transition frequencies between adjacent levels of the harmonic oscillator
are equal, then frequency indexes are omitted in Eq. (3.24).
First, we shall consider the case when the oscillator frequency is not
exactly equal to that of the field, i.e., f 4: to. Let us transform Eq. (3.24)
to)t]. For the new probability
using the substitution fit,
at, exp[iv(f
amplitudes fit, which differ from av by exponential phase factors one has
the following set of equations with constant coefficients:
dt
iv(O
to),= il[Vl/2[tv_
q"
(V
q"
1)1/2fly
+
1]
(3.25)
According to the general theory of linear differential equations, solutions
of (3.25) may be written in the form
fit,
ott,
(3.26)
exp(- iet)
Now for o, we have the set of equations
--[E;
+ L(’
to)]O/,
[vl/20/.v_
+
L
+ 1)l/20v
+
1],
(3.27)
and it can be seen that eigenvalues of e, i.e., such values which are
consistent with nontrivial solutions of (3.27), are just the quasienergies.
However, at first sight it is not clear yet what is to be called eigenvalues
in our case, since the set (3.27) is infinite. Indeed, with an arbitrary e,
const one can find c1 from the first equation,
setting, for example, ao
et2 from the second equation, etc. This procedure may go on to infinity.
However, from normalization requirements it becomes obvious that one
must choose only such solutions which, in any case, comply with the
condition
lim a,
0
(3.28)
The values of e, for which the condition (3.28) is met, will give just the
quasienergy spectrum. Here one may see the full analogy with the way of
finding the discrete eigenvalues of energy in stationary quantum systems,
where the requirement is for the wavefunction to tend to zero at infinity.
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
15
Thus, the common procedure to obtain solutions is rather clear. This
does not mean yet that they can be written out explicitly in a closed form.
Fortunately, in our specific case the chain of equations leads to rather
simple expressions for cry through the so-called Laguerre polynomials, the
properties of which are well known:
eo
Const x
where y
e/(f
polynomials are
LY
(x)
(v!)l/2 Ly
O-to
/
to)
+ ,2/(-
(3.29)
0))2. In the explicit form, the Laguerre
2
y (-x)
(-x)
y(y- 1)(-x)
+
+
+
v!
1! (v
2!
1)!
(v
2)!
y(y
1)’... "(y- v
v!
+ 1)
(3.30)
and one can make sure through direct substitution that ot (3.29) really
satisfies Eq. (3.27). Furthermore, from the explicit form of Laguerre polynomials (3.30) it can be established that the requirement (3.28) is fulfilled
if only the parameter y is equal to zero or to any positive integer. Therefore,
we obtain the quasienergy spectrum
e
-v(f- )
+
2
(3.31)
The numeration in (3.31) has already been given in such a form that as
the field amplitude (or the "y value) tends to zero the v-th dressed state
comes to the v-th energy level.
In particular, the dressed state with the quasienergy eo 2/(- o)
is adiabatically connected with the ground state of the oscillator, and,
according to the general principles, during adiabatically slow switching on
of the field the initially nonexcited oscillator has to be just in this dressed
state. In this case, the distribution of populations over various levels, the
normalization in (3.29) being taken into account, is given by the following
equation:
exp
I-(f
(3.32)
16
V. N. BAGRATASHVILI ET AL.
This is the well-known Poisson distribution with the mean energy
(3.33)
expressed in terms of the energy of one oscillator quantum. So, one can
conclude that effective excitation of the v-th level of the harmonic oscillator
under slow switching on of the frequency-detuned monochromatic field
requires that the ratio of the Rabi frequency /at the transition 0--* to
the value
to of detuning should be no less than v 1/2. The criterion
of adiabaticity in this case is the same as the condition (3.23) for a twolevel system.
In the case of stepwise switching on of the field, the dependence of
wavefunction of oscillator, or, in other words, the dynamics of excitation
is described by a superposition of dressed states. We omit here simple
intermediate calculations and write out at once the result in terms of populations depending on time:
(v(t))
pvv(t)
v.t
exp [- (v(t))],
(3.34)
where the mean energy of the oscillator (v(t)) in one-quantum-energy units
is
(v(t))
4/2
(
to)2
sinZI( to)t]
(3.35)
It should be noted that the expression for mean energy accurately coincides
with the corresponding classical result. It follows from Eqs. (3.34) and
(3.35) that the maximum of distribution oscillates in time with the freto between the ground state and the levels with v
)max
quency 1
4/2/([’ (0) 2.
Eq. (3.35) enables us to trace what occurs if f to, i.e., in the exactly
resonant case. As in the case of the classical oscillator in an exactly resonant
external field, the mean energy grows without limit according to the law
(v(t))
/2t2
(3.36)
Hence one can estimate, for example, that, with a dipole moment of the
lower transition of 0.3 Debye, the harmonic oscillator is excited by field
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
17
107 during the time of
with intensity 10 7 W/cm2 to levels with v
2
100 ns, i.e., absorbs an enormous energy. The possibility of such strong
excitation is caused only by the equidistant energy level spectrum of the
harmonic oscillator. This feature is also typical for some other quantum
systems with similarly equispaced energy levels in their spectra. 185 The
nonequidistanced levels of real spectra caused by anharmonicity for molecular vibrations bring about drastic changes in estimating the efficiency
of excitation. This can be seen if one estimates, with the above parameters
of oscillator and field, the value of detuning which leads to Vmax 1. Such
an estimate gives rather a small value 12
001 0.44 cm-1. Since the
typical values of anharmonic shifts in molecular spectra are larger, it
becomes clear to what extent the harmonic oscillator model is not adequate
to the treatment of an individual vibrational mode in view of its interaction
with radiation.
3.2.3. Multiphoton transitions for multilevel system
Three-level system. It is convenient to begin the examination of multiphoton resonant processes by discussing the dynamics of excitation of
the three-level system shown in Figure 3.2. We consider the case when
the frequencies of two successive transitions are not equal, i.e., 001o 4:
0021. From estimates for the two-level system obtained in Section 3.1 one
may expect that with, for example, the field frequency coinciding with
00o, the excitation of the highest level is probably effective if the value
2 /2 of power broadening for the transition
2 is greater than or
of the order of the corresponding detuning 1
002
10010 0021
In the opposite case when
I.
the oscillations must be observed only between two lower levels and the
upper level will be little populated. It is interesting, however, to examine
what will be observed if the field frequency is taken to be
001o
+
2
0021
(3.38)
chosen so that the excitation of the upper level from the lower one should
be resonant in energy with respect to the process of absorption of two
18
V. N. BAGRATASHVILI ET AL.
FIGURE 3.2 Two-photon resonance in a nonequidistant three-level system.
photons at once. Eq. (3.4) for probability amplitudes in this case has the
following form:
dao
--ilaexp( it21(i ’--t21t)
+i112a22 (it’ t2’)t2
da2
(339).
ill2alexp( it21-tlt)2
da
i/olao exp
exp
the rotating-wave approximation being used.
Let us introduce here the designation
tO,o)/2 12 Oo
((D21
and make in Eqs. (3.39) the substitution E1
a exp (it). As a result,
one gets equations with constant coefficients and after the substitution
ao,2
to,2 e-iet,
tl
Otl e-iet
(3.40)
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
--
19
gets the following set of linear equations for Ok:
EO/. 0
010/.1
0
(E
)1
010
120/.1
""
"-
0
EO/. 2
122 --0
(3.41)
which is intended for finding the eigenvalues of quasienergy 8i(i
0, 1,
2) through the requirement that its determinant be equal to zero. This
requirement leads to the cubic equation
E3
"" al2
(gl
-]-
2) E
0,
(3.42)
the roots of which in our specific case are
EO
O,
El,2
-{1 [1 +
+_
(3.43)
Now we have everything to write out the solutions for probability amplitudes corresponding to each dressed state and make up their superposition which describes the dynamics of excitation of a system being initially
at
0 in the lower level. First, let us assume that /01
/and
/12
use the condition (3.37). Neglecting the terms of the order of //g one finds
that the system oscillates between the lower and upper levels according to
ao(t)--exp(-i-t) cos-t,
a2(t)
-iexp(-i---t)sin-t
(3.44)
whereas the probability amplitude of the intermediate level is small (about
//15). Hence we have for populations
2
Poo(t)
2
COS2-t, P 22(/) sin2-/
(3.45)
By analogy with the two-level system, the value /2/I g is called the
two-photon Rabi frequency /2), and further we have to discuss the way
for the obtained expression to be generalized to the case when Ol 4: /12,
i.e., dipole moments of two transitions are not equal. One may assume
two following simple formulas which generalize the result just obtained:
20
V. N. BAGRATASHVILI ET AL.
/01 12
,(2)
(3.46)
and
201
y2)
-}-
212
(3 47)
The second equation (3.47) is formally true for the case of exact twophoton resonance (see the discussion in Ref. 185). Indeed, if with the use
of quasienergy values (3.43) one writes out explicitly, for example, the
dependence of the probability amplitude of the upper level on time, the
following equation will be found:
P22(t)
4y2y22
(y02
y122) 2
_.
sin2
/ +
28
/22t
(3.48)
However, we may notice that the population of the upper level in this
case does not reach unity at any instant. Moreover, if the dipole moments
of two transitions differ greatly, then the population of the upper level
should be much smaller than unity. The cause of this oscillation of populations is that at different dipole moments of two transitions the frequency
of exact two-photon resonance is not optimum for the excitation of the
2 separately
upper level. Let us consider the transitions 0--+ and
(see Figure 3.3). As discussed in Section 3.1 (Figure 3.1), the monochromatic field detuned from the resonance is responsible for shifts of
levels due to the dynamic Stark effect. It may be seen that the intermediate
level, detunings being equal in value but different in sign (such detunings
are at the exact two-photon resonance), and dipole moments of transitions
being different, affects the shifts of lower and upper levels in different
ways. Therefore, the frequency optimum for the two-photon transition
effectively shifts. The shift of the frequency "opt away from (tOo + 0)21)/
2 is small, and, in our case when I/Ol I, 112 I1 I, it will be
-
-opt
OJlO q- 0)21
y122- Y)I
2
28
(3.49)
It is interesting to point out that, in the general case, this shift depends on
the field amplitude. 186
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
21
FIGURE 3.3 The effect of different shifts of levels due to the dynamic Stark effect on the
value of frequency which is optimum for the two-photon transition.
More accurate analysis shows that, when the field at the frequency -opt
is applied, the system really oscillates between the lower and upper levels.
As far as the value of two-photon Rabi frequency is concerned, one should
now use Eq. (3.46) at the frequency Oopt. Thus, during the excitation at
l’opt the dynamics of populations of the lower and upper levels is described
by the following equations:
Doo(t)
COS
2/01
12_
t,
DEE(I)
sin2/o
’
12
(3.50)
To modify more general equations (3.19) which involve the detuning
from resonance to the case of a two-photon transition one should simply
use /(2 (3.46) instead of /and write the detuning in the form 22 2opt,
since it really gives the difference between the energy of two photons
divided by h and the frequency of transition. As a result, we get the
following equations:
22
P00(t)
PEE(t)
V. N. BAGRATASHVILI ET AL.
-
cos 2
(/
]
’Pt)2 "+" "01"
2 J
12
(3.51)
(a a opt) 2
sin2 (a
aopt) 2 + o/1
’2
0112/2 + (- opt) 2
0112/
(a aopt)2 + 0112q
2 2
2
+ (- opt)
82 J
o12/8
2
2
2sin2f
1/2t
which become Eqs. (3.50) when I
opt. The half-width of the twophoton resonance, or in other words, the value of two-photon power broadening is just equal to ,(2) (3.46), i.e.,
(A,-)(2)
,,(2)
(3.52)
It should be noted that, comparing the second Eq. (3.51) with the exact
solution (3.48) describing the excitation at the frequency f
(O1o +
to21)/2, one may find them to just coincide when our estimate (3.49) is
taken for the frequency fopt.
N-level system. The generalization of obtained derivations for cases
of resonances of greater multiplicity (three-photon, four-photon, etc.) looks
rather simple, is5 Let, for example, one intermediate level coupled with
the lower and upper levels through allowed transitions fall on each quantum
of radiation frequency in a one to one correspondence (see Figure 3.4)
until the N-th level is in the N-photon resonance with the ground one, i.e.,
the frequency of field is approximately equal to (tOo + tO21 +
mv-1)/N with the correction arising from shifts due to the dynamic Stark
effect. Then the multiphoton Rabi frequency is
""
(3.53)
(12
tOo)(21"
(.olO
o21)"... "[(N
1)D,
(DIO
(021
{,.0N_I,N_2]
For this expression to be valid, it is necessary, of course, that any intermediate resonance be absent, i.e., for all intermediate transitions, multiphoton Rabi frequencies should be much smaller than corresponding de-
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
23
N/,/
2
0
FIGURE 3.4 N-photon resonance in the quasiequidistant system with (N + 1) levels.
tunings. With this reservation, the dynamics of populations at multiphoton
resonance is described by the equations
Ooo (t)
cos2[/(mt],
prr(t)
sin2[/(t],
(3.54)
and the half-width of the multiphoton resonance is
AI’) (
2l(U)/N
(3.55)
Particular estimates for rates of multiphoton transitions will be given in
the next section when the anharmonic oscillator will be considered, for
which multiphoton resonances are of special importance. Here we only
wish to discuss two more important points. The above formulas for multiphoton Rabi frequencies can be applied to the simplest case when one
intermediate level falls on or corresponds to each quantum of field frequency. In the general case, there are perhaps several channels (see Figure
3.5a), and one should take into account their interference. Summing must
24
V. N. BAGRATASHVILI ET AL.
"-
(-Jfo (’Jzo-
"
IGURE 3.5 (a) Interference of several channels at the multiphoton resonance, (b) The
example of two-photon resonance with the Rabi frequency equal to zero because of the
interference.
be carded out with the use of expressions like (3.53), but with some
allowance made for the signs of transition dipole moments as well as
detunings. Figure 3.5b shows a simple example of a two-photon transition
with two intermediate levels with opposite detunings. The reader may
practice solving the equations for probability amplitudes and make sure
that, in this case, two-photon transitions do not really occur. The formal
summation of two-photon Rabi frequencies with sign-opposite detunings
at two intermediate levels also leads to zero.
Thus, the interference of various channels may cause the rate of multiphoton transition not to increase obviously but probably to decrease as
compared to the rate calculated for any separate channel.
The second important point concerns the possible role of the nonmonochromaticity of radiation. Here the case is just as that for the two-level
system. In particular, only the resonant part of field spectrum may play a
role, and one must take into account this point when estimating rates of
multiphoton transitions. It should be noted, however, that multiphoton
Rabi frequencies are ordinarily much smaller than Rabi frequencies in the
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
25
two-level case. Therefore, the part of the field spectrum which may be
treated as resonant is much narrower. The characteristic estimates will be
seen from the instances considered in the next section.
3.2.4. Anharmonic oscillator
To allow for anharmonicity in a separate vibrational mode is the next step
after the harmonic approximation, when developing the theory of vibrational spectra. As noted in Ref. 175 (Section 2.1.3), a convenient approximation used here is the Morse oscillator, the energy levels of which
are given by Eq. (2.23) from Ref. 175. We do not write out the equations
of motion for probability amplitudes in this case, since these equations
cannot be solved analytically. However, we already know enough to understand the basic feature regularities of the dynamics of the anharmonic
oscillator depending on the frequency and amplitude of the exciting field.
If the frequency of field coincides with that of the transition v
0---> v
of oscillator, and the condition (3.37) is fulfilled, then population
oscillations between principally the pair of lower levels will occur, and
the excitation of even the next level v
2 will be small. Otherwise, if
the field frequency is shifted to the long-wave region, then successive
multiphoton resonances will be observed at the frequencies approximately
equal to tOlo Ixl, o,o 21xl, etc., where x is the anharmonicity constant.
From the equations derived in the foregoing section it is easy to obtain,
neglecting small anharmonic corrections for the dipole moments of transitions, the following expressions for multiphoton Rabi frequencies in the
case of a Morse oscillator:
,yN)
N1/2O1
1)!]3’Zlxl-I
[(N-
(3.56)
The typical dependencies of multiphoton Rabi frequencies (N 2-5) on
the intensity of monochromatic radiation are shown in Figure 3.6. The
oscillator parameters are given in the caption to this figure. It follows from
the results of the foregoing section that from the values of the multiphoton
Rabi frequencies one can estimate the half-widths of multiphoton resonances using Eq. (3.55). If for example, the anharmonicity constant Ixl
2 cm-1, the radiation intensity is 107 W/cm2, and the dipole moment of
the transition v
0---> v
is 0.3 Debye, the half-width of the threephoton resonance is just 10-3 cm-1 Thus, it may be seen that the note
made at the end of the foregoing section on the role of the nonmonochroma-
26
V. N. BAGRATASHVILI ET AL.
FIGURE 3.6 The dependencies of multiphoton RaN frequencies on the radiation intensity
at the following parameters of anharmonic oscillator: I.o
0.3 Debye; (a) Ixl
2
cm-’; (b) Ixl 5 cm-1.
ticity of laser radiation is really very essential for estimating the times
required for these multiphoton transitions.
Now let us consider the dependence of the population Pvv of the v-th
oscillator level on the field frequency, particularly averaged over a long
period of time. At the frequency of the v-photon resonance the value of
pv(t) will be maximum and approximately equal to 1/2, but peaks in
p(t) will be observed at the frequencies of other resonances too. An
example illustrating this is given in Figure 3.7. This figure also shows the
effect of the broadening of resonances as the field intensity increases.
When the field is so strong that the resonance half-width exceeds the
anharmonicity constant, i.e.,
Ixl
(3.57)
the adjacent resonances become overlapped, and the frequency of multiphoton resonance is not yet distinguished. The condition (3.57), as the
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
27
FIGURE 3.7 The dependence of the time-averaged population of third anharmonic-oscillator
level on the exciting frequency. 187 The parameter values are: curve 1: /Ol/21x 0.1’ curve
2:
lo/21x
0.4.
field amplitude grows, is not reached for all resonances simultaneously.
Designating as Nmax that maximum value of N, for which the condition
(3.57) is just met at a given intensity, one often refers to the range of
radiation frequencies
> tOo Nmax as the classical region. This
means that in this frequency range the quantum features of the oscillator
are only slightly important, and its behavior is close to the classical case.
The promotion of a classical nonlinear oscillator by an external monochromatic field is a problem treated in many textbooks (see, for example,
Ref. 188). The classical limit, however, is not of particular interest to us
since, in the following experiments, the laser radiation intensity is not
usually higher than 108-109 W/cm2. At the same time, for the oscillator
2
parameters used for Figure 3.6, the equality, for example, (Aft) (3
cm is reached at the intensity of radiation 1.6
109 W/cm2.
For the vibration-rotation molecular spectra, the sequence of transitions
corresponding to the Morse oscillator may be approximately realized if
Ixl
-
28
V. N. BAGRATASHVILI ET AL.
one, for example, limits himself to choosing the transitions without a
change in rotational quantum numbers. But it is easy to show that the
frequencies make up an arithmetic progression for successive transitions
within any given rotational branch, too. Let us take, for example, the
transitions within the P- or R-branches of spherical tops [see Eqs. (2.49)
from Ref. 175], or within those of parallel bands of symmetrical tops [see
Eqs. (2.51) from Ref. 175]. It may be seen that, neglecting the small
change in the rotational constant, one can formally introduce the effective
anharmonicity constant Ixl B. Thus, if one considers the P- or R-branch
separately, the anharmonicity constant apparently decreases. 189,]9 However, usually B
Ixl for molecules and the decrease has no particular
effect on estimates. Otherwise, if one considers sequences of transitions
in different bands, a wide variety of multiphoton resonances become possible. We shall turn to the discussion of this point in Section 5 of our next
paper, where the role of some other effects will be discussed which arise
from splittings of vibration-rotation states [see Ref. 175 (Sections 2.1.5
and 2.2.4)]. These splittings complicate the spectroscopic picture and,
hence, make it necessary to consider more complex multilevel systems
than those concerned in this Section.
,
3.3. Incoherent excitation of quantum systems with relaxation
In an analysis of the processes of absorption and stimulated emission, one
may often hear the term transition rate. This value is expressed in s-1 and,
in the case of a two-level system, is logically used to describe the dynamics
of populations of the lower and upper levels in accordance with the following equations:
dpoo
dt
dpl
dt
W(Ooo
O)
(3.58)
W(poo
p)
usually referred to as kinetic equations. It is trivial to solve Eqs. (3.58).
If, for example, initially at
0 the system is in the lower state, the
solution has the form
Poo
[1
+ exp(-2Wt)],
PI
[1
exp(-2Wt)],
(3.59)
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
29
i.e., the transition is saturated for the characteristic time Tsa
W. However, the analysis carried out in Section 3.1 of the behavior of the twolevel system in a monochromatic field shows that it is hardly possible to
describe the dynamics of coherent excitation in terms of transition rates.
In particular, at short times, when the population of the upper level is
negligible, it grows [see Eq. (3.19)] proportionally to 2, whereas the second
of Eqs. (3.59) gives 911
t. Also, according to (3.59), the populations
do not exhibit any oscillations but exponentially reach their mean values
1/2.
3.3.1. Conditions
of incoherent excitation for two-level system
The Eqs. (3.58) which are also referred to as the incoherent approximation
are often a good approximation in the presence of some factors which lead
to the broadening of spectra. In Ref. 175 (Section 2.6) we have already
treated the effects responsible for spectral line broadening. Getting ahead,
we may present a criterion when Eqs. (3.58) work well. This criterion
requires that the line-width be much larger than the value of power broadening. In this section we shall treat the effect of both inhomogeneous
(Doppler) and homogeneous (collisional) broadening. One more important
case when kinetic equations may be used should also be pointed out. This
is the situation when transitions into many close states are possible from
the initial level, and it will be discussed in Section 3.7.
Let us begin with the excitation of a gas of two-level non-interacting
molecules, with statistically distributed velocities, i.e., having different
transition frequencies due to the Doppler effect. We shall be interested in
the dependence on time of the excitation probability of the upper level
averaged over all molecules. With the use of Eqs. (3.19) and (2.119) (from
Ref. 175) this dependence can be easily written explicitly:
(O,,(t))
sin_
I_[(fl-
2,rrkT
o)2 +
4,
s
4,7,]’/2t} exp[
(l
to)
+ 41o,
M(D,- to)c
ff
]/ao,,
(3.60)
where, for simplicity, the field frequency is taken to coincide with the
center of the Doppler contour. The integral entering Eq. (3.60) contains
two functions which depend on frequency and have their maximum at that
of the field. In the cases when the width of these functions differ greatly,
30
V. N. BAGRATASHVILI ET AL.
the variation of the broader function of frequency may be neglected, and
we may pull it outside the integral.
If the Doppler half-width is much smaller than the value 21- o 1 of power
broadening the average population of the upper level behaves, of course,
almost according to Eq. (3.19). But now we are only interested in the
opposite case when
(A0))Dppl
C
M
In2
>> 21 o,I
(3.61)
In this case, the integral in Eq. (3.60) can be expressed through an integral
of the zero-order Bessel function which, in its turn, can be written conveniently as the series"
(3.62)
The derived equation shows that, at small times, the average population
of the upper levels now grows proportionally to time as given by Eqs.
(3.58) with
W
kT ]
-\f 27rM
c
1/2
11,
(3.63)
though the population dynamics is not described by Eqs. (3.58) within the
whole time interval.
The case of inhomogeneous Doppler broadening considered is of little
interest for the goals of this series of papers but is illustrative. Indeed,
with the radiation frequency 1000 cm-1 and the transition dipole moment
0.3 Debye, the condition (3.61) is realized only at very low radiation
intensities < 10 W/cm2. For our purposes, it is more important to examine
the role of collisional broadening, since in our discussions we shall come
31
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
across some experiments in which the collisions between molecules in a
gas have time to occur during the laser pulse. Besides, in the case of
collisional broadening and at sufficiently frequent collisions, Eqs. (3.58)
prove to be valid now within the whole time interval.
It is not, however, quite apparent how the collisional relaxation processes
can be included, since so far we dealt with the Schr6dinger equation or
with the equations of motion for probability amplitudes only in view of
isolated quantum systems. We have noted in Section 2.5.7 from Ref. 175
that the description using the density matrix is the most general statistical
one. The formalism of a density matrix is well adapted to the description
of relaxation processes. On the other hand, the equations for probability
amplitudes may always be transformed into those for elements of the
density matrix. In the case of a two-level system in particular, one can
and Pll
obtain the equations for the diagonal elements poo
a*al
aao
of the density matrix as well as for nondiagonal ones pol
aal and
starting from Eqs. (3.11). These equations of motion are
aao
dpoo
dt
dt
d9ol
i1ol {pol exp[i(f
i’ol {pol exp[i(f
dt
ilol (Poo
91o
9ol
to)t]
to)t]
911)exp[-i(f
plo
exp[-i(l
131o exp[- i(fl
plo
to)t]}
to)t]}
(3.64)
to)t]
The solutions of Eqs. (3.64) are, of course, absolutely identical to those
of Eqs. (3.11). But Eqs. (3.64) are more convenient in view of the natural
inclusion of terms which describe entering and leaving the lower and upper
levels of the working transition due to the relaxation processes. Such terms
should be introduced into equations for the diagonal as well as nondiagonal
elements of the density matrix. In addition, a term of type (2.117) (see
Ref. 175) enters the equations for nondiagonal elements which describes
their relaxation to the zero equilibrium value (see the discussion in Section
2.5.7 from Ref. 175). We do not write out here the general equations
which are usually referred to as the Bloch equations. TM In the general
case, the solutions of these equations are rather cumbersome, and they are
studied in some manuals (see, for example, Refs. 176 and 180). However,
here is another point of interest for us. Let us look at the kinetic equations
(3.58) for the populations of levels. If one assumes that these equations
V. N. BAGRATASHVILI ET AL.
32
are a good approximation, then it is not difficult, in principle, to include
the relaxation terms into them. The only process which cannot be included
directly into Eqs. (3.58) is the phase or transversal relaxation of the densitymatrix nondiagonal elements. Therefore, it is appropriate to ask in which
cases the solutions of the equations for density-matrix elements, in the
presence of phase relaxation only, are close to the solution (3.59) of kinetic
equations.
Thus, let us introduce into Eqs. (3.64) the terms which describe relaxation of nondiagonal elements to zero and suggest for simplicity that the
field frequency is equal to the transition one. We come to
dpoo
dt
d011
dt
dOo
dt
dOl o
dt
i’/Ol(O01
i’YOI(P01
OlO)
O10)
(3.65)
i3’ol(POO
i’/oI(PO0
O11)
Pl 1)
T2OO1
,"PlO
12
We are interested in the solution of these equations when the system is
0 in the lower state, i.e., Ooo(t
1, Oll(t 0)
0)
initially at
Ool(t 0)
Olo(t 0) 0. The solution can be found easily and for
the diagonal elements of the density matrix, in particular, it has the following form:
(3.66)
Oil
lexp(22) (ch[(1- lt/Ol 2]
2
2
2
(1- 16To21T2) 1/2sh
(1-
Tro\l/2
2)
16To21T2)I/22
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
33
The behavior of this obtained solution is different depending on the
relation between the values 4/o and T-1. With the condition 41 o 1 >
T-1, the population oscillations are observed which decay with the characteristic time 2T2, the populations tending to their average values equal
to 1/2. If the opposite condition 41o1 < T-’ is fulfilled, the equilibration
of populations occurs without oscillations, and this process is described
by two exponents. Finally, if 41"o,I T-’, then only one exponent plays
a significant role:
Poo
1+
1
exp (-4/o2T2t),
Pal
2
1
exp(-4/21T2t)
2
(3.67)
-
Comparing Eqs. (3.67) with Eqs. (3.59) one may conclude that the solution
of density-matrix equations in case 41"Oll
T coincides with that of
kinetic equations, if
W
2/T2
(3.68)
is taken. Thus, sufficiently frequent phase-interrupting collisions destroy
the coherence of interaction of the two-level system with the field.
In Section 2.6 from our previous paper (Ref. 175), when discussing
spectral-line broadenings, we introduced the value of an absorption crosssection [see Eq. (2.122) from Ref. 175]. Now one may correlate the
transition rate W with the absorption cross-section. The cross-section includes the factor Ga which gives the line shape. In the considered specific
cases when the field frequency lies in the absorption contour center, the
factor Ga can be found from formulas from Ref. 175, Eq. (2.119) for an
inhomogeneously broadened line and Eq. (2.121) with (mto)hom
Tf
for a homogeneously broadened line, if, in these equations, to
too is
taken. It is easy to obtain from Eqs. (3.63) and (3.68) that, for both the
cases considered,
-
W-
hf
crP,
(3.69)
where P denotes the radiation intensity expressed in terms of photon x
cm-2 x s This formula for the transition rate remains valid for the more
general case, too, when the field frequency is shifted from the center of
the absorption contour, but one should not forget that the concept of a
-.
34
V. N. BAGRATASHVILI ET AL.
transition rate itself makes sense provided that rather moderate fields are
used for excitation.
From the analysis carded out, one important conclusion can be drawn.
Whereas, under coherent excitation, the characteristic transition time is
inversely proportional to the electric component of field, this time is inversely proportional to the intensity of radiation under the incoherent excitation. Accordingly, for the saturation of a transition under incoherent
excitation, the fluence of the excitation wave is of importance, and the
characteristic required value (I)sa
hfll2r is called the saturation energy.
3.3.2.
Effect of rotational bottleneck
In Section 2.5 from Ref. 175, when discussing collisional relaxation processes in gases, we distinguished the rotational relaxation as the fastest of
the processes which change populations of vibration-rotation molecular
levels. In this section we want to treat the kinetics of excitation of a quasitwo-level vibrational transition in the presence of rotational relaxation,
taking into consideration the real distribution of molecules over the rotational sublevels of the ground vibrational state. For simplicity, we shall
neglect the initial population of the upper vibratational state and assume
that the radiation field effectively interacts with only one vibration-rotation
transition, i.e., with a really small fraction of all molecules in the gas.
The last assumption actually means that the value of power broadening is
much less than the characteristic difference of frequencies of two adjacent
transitions in the vibration-rotation spectrum. But, as will be seen below,
this assumption is not absolutely necessary.
Thus, we consider (see Figure 3.8) the field frequency to be resonant
to one of the transitions in the vibration-rotation spectrum and assume that
the direct excitation from the rest of rotational sublevels may be neglected.
The molecules originally in non-resonant levels are allowed to be excited
effectively only if they enter the resonant level by the process of rotational
relaxation. Now let us treat what the equations are for the adequate description of the kinetics of the process. Since the very sense of the problem
makes for insight into the behavior of a system during time intervals longer
than the rotational-relaxation time, it is natural to use the kinetic equations
for the populations of levels. Indeed, the rotational-relaxation time is at
least no less than the time T2 of the phase relaxation. If the value of power
broadening is much smaller than Tf 1, the kinetic equations, as has been
shown in the foregoing section, give a good approximation for describing
the excitation of the working vibration-rotation transition. On the other
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
35
FIGURE 3.8 The role of rotational relaxation in the process of excitation of molecular gas
when the field interacts with one vibration-rotation transition.
hand, any other ratio between the value of power broadening and the phase
relaxation time forces the populations of working levels, as follows from
Eqs. (3.66), to tend toward their equilibrium values 1/2 during a time about
T2, i.e., the use of kinetic equations which, for this case, is not quite
correct physically should not, nevertheless, lead to any essential error.
Now let us discuss a possible way to describe the rotational relaxation
in our specific situation. In Section 2.5.3 of Ref. 175 we discussed various
approximations to describe relaxation processes phenomenologically. For
our case, the simplest linear approximation is quite suitable. The arguments
for using the linear approximation are rather simple. The radiation, as it
acts on one vibration-rotation transition, is able to create strong nonequilibria only in active vibration-rotation levels. (According to Figure 3.8,
we shall designate their populations as No and N1.) Since the fraction of
molecules interacting with the radiation is small, the populations of the
rest of the levels differ just slightly from their equilibrium values, and this
fact justifies the description of their populations within Eqs. (2.109) from
Ref. 175.
Further, let us consider one of the vibrational states, say the lower one,
and designate the populations of sublevels which do not interact with
radiation as Nj and the total population of the lower vibrational state as
/o. The equilibrium populations of different rotational sublevels can be
represented as F.o, where Fj are the Boltzmann factors (see Section 2.5.1
from Ref. 175). Within the linear approximation, the relaxation term for
each rotational sublevel may be written out in the following form:
36
V. N. BAGRATASHVILI ET AL.
F’Io
dN
dt
N
(3.70)
rot
On the other hand, the process of rotational relaxation cannot change the
total population of the vibrational state, i.e.,
No +
-
(3.71)
0,
dt
and, if one assumes that the rotational-relaxation rates are approximately
equal for all the really populated rotational sublevels, then the following
sequence of equations will occur:
dNo
t
d
d-t N
j
Trot
[(1
Trot
Fo)o
,
F .fVo
N
(3.72)
j
(o
Foo
No)]
Trot
No,
i.e., the same relaxation equation as (3.70) must be valid for a distinguished
sublevel which is perhaps strongly perturbed.
Now it is not difficult to write out the complete set of equations which
includes the interaction of radiation with the active transition as well as
the process of rotational relaxation. This is the set of four equations for
two populations No and N1 of the active vibration-rotation sublevels and
two total populations No and/1 of vibrational states. Of course, the equations for total populations do not include relaxation terms, and the full set
of equations has the form
dNo
dt
dN
dt
dt
dV1
dt
-crP(No
trP(No
o’P(No
trP(No
NI)
N1)
+
N)
NI)
+
Fo
No
Trot
FI
N1
Tro
(3.73)
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
37
where, for simplicity, the rotational relaxation times as well as the population factors are taken to be equal for the lower and upper states, and tr
denotes the absorption cross section of the active vibration-rotation transition. We are not writing out here the general solution of Eqs. (3.73)
obtained in Ref. 192 assuming sequential onset of the radiation wave. This
solution, which can be expressed in terms of two exponents, is rather
cumbersome, but with F 1 it may be essentially simplified and, for total
populations
and/1 of vibrational states, leads to the following approximate expressions which resemble solution (3.59) of the simplest kinetic equations"
Ao
/o
where the
[1
"
/1
+ exp(-2wt)],
=
1
exp(- 2wt)],
(3.74)
effective vibrational excitation rate is
w
1
crPF
+ 2tyP’rrot
(3.75)
The value (2w)-1 gives the characteristic time ’l’sa of saturation of the
whole vibrational transition. At low radiation intensities when the condition
tYP’rrot < is fulfilled, we have
’i’sat.’
(2trPF)-1,
(3.76)
i.e., the value ’l’sa is 1/F times larger than the time of saturation of an
active vibration-rotation transition alone. As the intensity increases, the
time of saturation decreases and, with the condition o’P’rrot
fulfilled,
reaches its minimum value
min
el’sa
’l’rot/F
(3 77)
It is this last property putting the limit on the excitation rate that is usually
referred to as the effect of rotational bottleneck. Physically this effect is
rather obvious. The active transition is quickly saturated by high-intensity
radiation. During an interval of time about %ot the molecules having been
excited into the upper active vibration-rotation level relax to other rotational
sublevels, and a small new portion of molecules about F enters the lower
active one, from where it is quickly excited upwards, etc.
38
V. N. BAGRATASHVILI ET AL.
We assumed above that the radiation effectively interacts only with one
vibration-rotation transition. Actually, for complex vibration-rotation molecular spectra, the field with a finite spectral width can resonantly interact
with a group of transitions. For this case, the obtained equations remain
valid although the population factor should be treated in some different
way (to avoid confusion we shall use the notation f for it). Now, the value
f will denote the relative resulting population of all rotational sublevels
interacting with the radiation. The f value understood in this way cannot
be always easily estimated theoretically. From the analysis performed,
however, a rather simple method follows which enables us to determine
it experimentally. One should measure the saturation energy with a short
radiation pulse, the duration of which is much shorter than the rotationalrelaxation time, and also with a long radiation pulse, the duration of which
is much longer that the rotational-relaxation time. Then, it follows from
the discussion on Eq. (3.76), the f value can be obtained from the ratio
of the saturation energies in the two experiments. This method had been
realized in Ref. 193 for the molecule C2F3C1.
There is one more important point to be mentioned. For molecules with
closely spaced vibrational-rotational lines, even within our simplified quasitwo-level model, the factorfmust grow with radiation intensity, since the
value of power broadening increases. This effect, no doubt, becomes
important when the value of the power broadening is greater than the width
of the radiation spectrum (see the estimates at the end of Section 3.2.1).
In this case, one can roughly estimate the factor f as the ratio of the value
of power broadening to the half-width (Av)/2 of that rotational branch
where the radiation frequency lies, i.e.,
f 41-ol/(av)
(3.78)
The obtained expressions can be modified, of course, with this effect being
taken into account. But, this is hardly worthwhile, since the experiments
demonstrate convincingly that, as the molecules are acted upon by rather
strong laser fields even under collisionless conditions, an essentially larger
fraction of molecules is excited than that allowed by Eq. (3.78). We shall
return to discussing this problem more than once. One can probably gain
some insight from the parallel observation that the higher vibrational states
than the fundamental one are excited, too, i.e., under sufficiently strong
fields, the molecule no longer behaves as the quasi-two-level system examined in this section.
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
39
3.4. Excitation from a discrete level into a quasicontinuous
band
In Section 2.3.4 of our paper (Ref. 175), we discussed the features of the
spectrum of vibrational transitions in the region of stochasticity of vibrational motion. In this region, the spectrum becomes, in fact, quasicontinuous because of the enormous density of states and strong mixing of the
harmonic wavefunctions due to anharmonicity. In other words, many transitions become possible from a vibration-rotation state, their frequencies
being very close and covering a rather wide spectral range. It is evident
that, in the presence of a large number of transitions, the two-level approximation is invalid if the value of power broadening becomes comparable to or exceeds the typical differences between frequencies of adjacent transitions. And what is more, the description with the use of kinetic
equations proves to be a good approximation for this case. To demonstrate
this we shall treat in this section the dynamics of excitation from a discrete
state into a band of closely spaced levels (see Figure 3.9). Such a process
is an elementary act of excitation into the vibrational quasicontinuum by
radiation.
0
FIGURE 3.9 Transitions from a discrete level to a band of closely spaced states.
40
V. N. BAGRATASHVILI ET AL.
3.4.1. Derivation
of kinetic equations
The equations for the probability amplitudes of the lower level ao and the
levels in the band a lj have, within the rotating-wave approximation, the
following form:
dao
/jalj
dt
exp [i(f/
to)t]
(3.79)
i%ao exp [- i(f/
to)t]
Here to is the frequency of transition from the j-th level in the band to the
lower state, /j
01f[1,j /2h is the corresponding Rabi frequency,
and the indexes 0 and relating to the lower level and the whole manifold
of upper levels are omitted for brevity.
Eqs. (3.79) cannot, of course, be solved explicitly for arbitrary distributions of dipole-moment matrix elements and frequencies over various
transitions. We shall consider the only known example for which it is
possible to get an analytical solution. TM This is the case when the levels
in the band are equidistant, i.e.,
to
tOo
+ j/hp
(-
oo
< j < oo),
(3.80)
where 9-1 is the distance between adjacent levels, and when the dependence
of / on j is Lorentzian, i.e.,
Here
so that
"
,2
’l’l’h 2
is the half-width of the
d-
j2/2192
tanh(’trhp)
(3.81)
/ curve, and the normalization is chosen
,
/z,
(3.82)
i.e., the value /may be called the integral Rabi frequency.
As usual, we shall be interested in the solution of Eqs. (3.79) for the
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
41
probability amplitudes generated by stepwise switching on of the field,
and with the following initial conditions:
ao(t
O)
1,
alj(t
0)
(3.83)
0
As the general theory dictates, the solution we are interested in can be
obtained if one finds the dressed states and makes up their superposition
complying with the initial conditions (3.83). However, in the particular
case under consideration, it is simpler, first, to transform Eqs. (3.79) to
one integro-differential equation which includes only the probability amplitude ao(t). This equation can easily be derived if one formally integrates
Eqs. (3.79) for alj(t) and substitutes the result into the first equation. Doing
this one comes to the following equation:
d- +
ft
K(t
’r)
ao()K(t- )d
O,
o
(3.84)
] / exp[i(l
to)(t
a-)],
J
the initial conditions are
ao(t
0)
1,
using Eqs. (3.80) and (3.81) for
K(z)
K(z
2 exp[i(O
+ 2"rrhp)
toj
tOo)Z] [chSz
K(z)exp[i(l
---(t
and
/,
0)
the kernel comes to
tanh("rrhp)shz],
tOo)Z]
(3.85)
0,
0 < z < 2.trhp
(3.86)
A standard technique for solving equations like (3.84) is to use the
Laplace transformation. We will not trace here the intermediate calculations
and shall write out the solution in its final form which may be checked
directly through substituting. This solution has a rather complex structure,
and all the necessary equations are given in Table 3.1. The solution, of
course, depends on the detuning A fl tOo of the field frequency with
respect to the center of the Lorentzian contour, and it includes the roots
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
43
of the cubic equation written out in the fourth line of Table 3.1. Of physical
interest for us is the situation when the half-width of the transition spectrum
exceeds the distance between adjacent levels in the band, i.e., the condition
> 1/hp
(3.87)
is met. The solution comprises with this condition the small parameter O
1
tanh(’rrShp) 1, and, for the roots of the cubic equations, there
are valid and rather simple approximating formulas written out in the fifth
line of Table 3.1.
Let us now examine the behavior of the probability amplitude within
the time interval 0 < t < 2"rrhp when ao(t)
Ao(t) (see Table 3.1).
Neglecting the terms which are small by the parameter O, one comes to
the following equation:
,
ao(t)
r
+
iA
i
rl
erlt
r2
+
r2
+
r2
-
iA
e2,
(3.88)
rl
where rl,2
( iA)/2 +_ [( iA)2/4 ,2]1/2. If the field frequency
0,
coincides with that of the center of the transition contour, i.e., A
then the obtained equation is similar in structure to Eqs. (3.66) which
describe the dynamics of a two-level system in the presence of phase
relaxation. As in that case, if the value 2" exceeds the half-width of the
contour of transitions, then the damped oscillations will exhibit the dythe population
namics of system and, in the opposite case when 2y
of the lower level decays exponentially according to
,,
Poo
aoao
exp(-
2y 2
--t)
(3.89)
With the detuning of the field frequency from the center of the contour
of transitions, i.e., with A :# 0, two regimes are also possible, the damped
oscillations and the purely exponential decay. As the criterion of the purely
exponential decay, the condition
y
,18
iA[
(3.90)
44
V. N. BAGRATASHVILI ET AL.
serves, and, in this case,
Poo(t)
exp
2/2i
--2 +
(3.91)
A2
With the purely exponential decay of the population of the lower level one
probably wishes to present the transition rate
2y 2
8 .+ A 2
W
(3.92)
2
which is proportional to ,2 and, hence, the radiation intensity in terms of
the absorption cross-section at the frequency of field. By introducing the
integral square of the dipole moment i 2 we find that
W-
2h 2 g2 + A 2
1. cc
"n’[g2
+ (
to)2]
"
(3.93)
In Section 2.6 from Ref. 175 we introduced the absorption cross section
(2.122) (from Ref. 175) for broadened lines. In that case the transition
spectrum was continuous but it is discrete in the case under treatment.
(’
Nevertheless, it is easily seen that the value (4"n’l2/hc) {/"gl’[ 2
to)z]} would be just the natural generalization of the absorption crosssection for our quasicontinuous spectrum of transitions, if the substructure
8/’rr[8 z + (fl to)z] introduced.
were ignored and the line-shape Gn
It should also be noted that the transitions from a discrete level occur
into that range of the quasicontinuous spectrum which is resonant to the
field frequency. This is mandated by the energy-conservation law and,
from the uncertainty principle, the width of the excitation range within an
order of magnitude is
"-
AE-- h/W
(3.94)
This conclusion follows from rather general quantum-mechanical principles, but it can be drawn directly from the solution of Eqs. (3.79) too.
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
3.4.2. Validity
45
of the kinetic equations
So far we have considered the population dynamics of the lower level only
within the time interval 0 < < 2"trhp and concluded that, at comparatively
weak fields, this dynamics is well described by a kinetic equation like
dpoo
dt
rPPoo
(3.95)
Now we want to clear up when it is sufficient to consider only this time
interval. This obviously requires that the population of the lower level
essentially decay during the time of order 27rhp, i.e., the condition
lao(2rrhp)
,
3.96)
be valid. Rewriting the transition rate W (3.92) with the use of Eq. (3.81)
in terms of the Rabi frequency res near the resonance, one comes to the
following natural requirement
/res
> 1/hp,
(3.97)
i.e., as discussed at the beginning of this Section, the value of power
broadening evaluated at an individual near-resonant transition must exceed
the distance between adjacent levels in the band. The condition (3.97), in
addition to (3.90), serves as the second criterion for validity of the kinetic
equation (3.95). This condition causes, in fact, the discrete structure of
the transition spectrum to be unessential, and one actually may speak about
a continuity of spectrum.
We now turn to particular estimates. Let the integral dipole moment of
the level-band transitions
(3.98)
be 0.3 Debye. From the mechanisms of stochastization of vibrational
motion in molecules as discussed in Section 2.3 from Ref. 175 it is clear
that the band widths in the vibrational quasicontinuum are typically 10-100
46
V. N. BAGRATASHVILI ET AL.
cm-1. Therefore, the first condition (3.90) necesssary for the validity of
kinetic equations requires that the radiation intensity be less than 101 W/
cm2. The typical intensities of laser radiation used in the experiments
discussed in this series of papers are ordinarily much smaller than this
value.
The second criterion for using the kinetic equations put requirements
on both the lower limit of intensity and the density of states. One should
keep in mind that, with the integral Rabi frequency fixed, the value Yres
is inversely proportional to p 1/2. Let, for example, the field frequency lie
near the center of the Lorentzian contour of transitions. Then one may find
that the condition (3.97) transforms into
y2hp > ’n’8
(3.99)
Take for estimates 8 10 cm-1,
0.3 Debye, and the radiation intensity
107 W/cm2. Then the condition (3.99) is fulfilled when the density of states
exceeds 103 cm corresponding, for example, to the density of vibrational
states of the molecule SF6 at the energy --5000 cm-1. As will be seen
below, when discussing experimental results, the value of radiation intensity taken is rather typical for observing the excitation of molecules into
the vibrational quasicontinuum. On the other hand, the estimated value of
the state denisty seem also probable for the lower energy limits of stochasticity regions in molecules. From this it follows that the description
of the dynamics of excitation in the vibrational quasicontinuum within the
kinetic equations is perhaps a rather good approximation. Furthermore, it
should be added that the usual width of laser-radiation spectrum obviously
reduces the condition on the density of states required for using the kinetic
equations.
3.5. Excitation of the vibrational quasicontinuum
In the foregoing section we have established the conditions in which the
radiation field produces the exponential decay of a discrete level into a
quasicontinuous spectrum. The same conditions are required, in fact, for
using the kinetic equations in the case of a more complex system when
resonant transitions occur between two bands of the quasicontinuous spectrum (see Figure 3.10).
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
47
FIGURE 3.10 The relationship between the rates of upward and downward transitions
between two quasicontinuous bands.
3.5.1. Kinetic equations for stimulated transitions in vibrational
quasicontinuum
We shall not discuss here the substantiation of kinetic equations for this
case. (The reader may turn to Refs. 195-197.) In this section we shall use
the kinetic equations without any additional reservations, but first we want
to focus our attention on some important physical points.
If at the initial instant of time the system is at some level of, say, the
lower band, then the radiation produces transitions into a narrow part of
the upper band. As has been mentioned in the foregoing Section, the width
of this part is of the order of the inverse transition rate. Further, we should
also take into account the downward transitions which led to populating
the narrow part of the spectrum in the lower band. Therefore, thinking
about the populations which enter the kinetic equations one should consider
the total populations summed over the resonant parts of bands. (We shall
designate them as Zower and Zupper. On the other hand, the description of
48
V. N. BAGRATASHVILI ET AL.
the excitation dynamics in terms of total populations must presuppose that
all the levels are rather equivalent, i.e., the transition cross-sections for
them are approximately equal. In Section 2.3.4 from Ref. 175, when
discussing the spectra in the vibrational quasicontinuum, we noted that the
dipole moments of transitions, besides their regular average change with
the energy, according to Eqs. (2.83) from Ref. 175, may also undergo
irregular variations from transition to transition. It is easy to understand
(see, for example, Ref. 198) a rough criterion for when it is possible to
use the averaged dipole moments. It is necessary that the characteristic
spectral scale of dipole-moment fluctuations be smaller than the inverse
transition rate which fixes the width of the populated part of the spectrum.
We shall assume that this criterion is fulfilled. Then it is physically justified
to use the averaged value of i 2, through which one can naturally introduce
the cross-section, as was done in Section 2.6 of Ref. 175 as well as in the
foregoing one.
Another important point concerns the ratio between the rates of upward
and downward transitions. In Section 3.5, discussing the dynamics of
incoherent excitation of a two-level system, we came, in fact, to Eqs.
(3.58) which exhibit equal rates of upward and downward transitions. In
the foregoing Section while discussing the decay of a level into the quasicontinuous spectrum, we have shown that it is described by Eq. (3.95)
which does not include downward transitions. It may easily be understood
that these results reflect two specific ratios between state densities in considered cases, and they can be simply generalized for the situation (see
Figure 3.10) when the state densities are arbitrary. In the general case for
stimulated transitions between two bands of the quasicontinuum, the ratio
of the upward-transition rate W (lower-- upper) to the downward transition
one W (upper
lower) must be equal to the ratio Pupper/Plower between the
densities of states in the bands. Thus, one comes to the following equations
for populations:
dt
dZupper
dt
trP
rP
(
Zlowe
Zlowe
Pupper
Zupper
Plower
Zuppe
Pupper
(3.100)
)
where r means the absorption cross-section from any level of the lower
band, with the above reservation on the possibility of averaging this value.
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
49
We may now write out the kinetic equations for subsequent stimulated
transitions (see Figure 3.11) between many sectors of the quasicontinuum
one quantum of radiation apart. We shall mark the total populations z, of
the vicinities of narrow resonances as well as the corresponding densities
of states p. by the index n. The equations take the form
dt
l’n
l,n
(
P ZnO’n,n +
P(Zn
\
n+
Zn
+
(3.101)
These equations describe the dynamics of populations for given initial
conditions and, for insight into the features of appearing distributions, one
may test various explicit dependencies of the cross-section on the number
FIGURE 3.11
quasicontinuum.
Multistep transitions between narrow resonant sectors in the vibrational
50
V. N. BAGRATASHVILI ET AL.
of transitions which, of course, are of physical sense. It should be noted
fight away that one can distinctly see the analogy of Eq. (3.101) to those
which describe the problems of one-dimensional random walk in the probability theory (see, for example, Ref. 199). A standard approach to studying
such problems is the approximate procedure which comes to replacing the
set of ordinary differential equations by one partial differential equation
which is usually called the Focker-Planck equation.
If in Eq. (3.101) the rates of upward and downward transitions were
equal we would actually deal with the well-known process of one-dimensional diffusion. If initially the distribution is concentrated near the boundary corresponding, in our case for example, to the initial conditions
Zo(t
0)
1,
Zn(t
0)
0 for n >
(3.102)
then the diffusion leads to the qualitative shape of distributions shown in
Figure 3.12a. The maximum of distribution always falls on the boundary
and, for that particular case when the diffusivity (or the transition rate for
FIGURE 3.12 The qualitative shapes of distributions realized during processes like diffusion
when initially the distribution concentrates near the boundary: (a) diffusion without drift; (b)
diffusion plus drift.
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
51
us) is a constant value, the width of distribution grows proportionally to
1/2"
However, in Eq. (3.101) the rates of upward transitions are greater than
those of downward ones since the density of molecular states increases
versus the vibrational energy. Therefore, we may say that we soon deal
with a process which can be referred to as diffusion plus drift. In this case,
the qualitative shape of distribution is different, since the drift forces the
maximum to be shifted from the boundary (see Figure 3.12b).
3.5.2. Excitation for the case
of constant cross-sections
The difference in rates of upward and downward transitions is an important
principal point, and we can illustrate this through direct calculations suggesting, for example, that in Eq. (3.101)
O’n-
1,n
O"
const,
1/)n
Pn
-
0/,
const
<
(3.103)
To simplify the mathematics let us consider the infinite number of bands,
and Eq. (3.101), with the cross-sections and state densities chosen: take
the form
dzo
dt
azl)
-crP(zo
(3.104)
(1
rP[z._
+ e0zn + ezn
+
1],
<n<
The explicit solution of Eqs. (3.104) with the initial conditions (3.102) is
expressed through the infinite series of modified Bessel functions:
z.(t)
ol.
-n/2
+ 01.1/21
+
exp[ (1
+ a)rPt] {In(2ol/ZcrPt)
l(2ol/2crPt)
(1
om/21
or)
(3.105)
+2+
m(2a 1/2crPt)
0
The modified Bessel functions can be presented as the series
In(x)
k= 0
k!(nt’"’ + k)!’
(3.106)
52
V. N. BAGRATASHVILI ET AL.
and, using the elementary relations (see, for example, Ref. 200)
di
dIo
I,,_ (x) + I,,
2-7"
+
(x),-;-
l(x),
(3.107)
one may convince himself through the direct substitution that Eq. (3.105)
actually gives the solution required. Though the solution (3.105) looks
rather cumbersome, it is easy to estimate such very important characteristics
of distribution as the mean n and the relative half-width d
n2
The
Ref.
from
n 2
equation
1)1/2 [compare with Eq. (2.98)
175].
for n can be found if one multiplies Eqs. (3.104) by n and carries out
the summation over n using the trivial equality
Z
(3.108)
2n(t)
0
This equation has the form
d(n)
dt
dt
’
ngn
(1
a)rP + arPzo,
(3.109)
hence it follows that
(n)
(1
a)rPt + arP
Zo(r)d
(3.110)
o
Right away one may see a principal difference between the cases when
1, the first linear term inEq. (3.110)
and ct < 1. With ct
vanishes, and using Eq. (3.105) we get
ct
n
trP
exp(-2trP’r) [lo(2trP’r)
+ ll(2rPr)]d (3.111)
o
At large values of the variable the modified Bessel functions entering Eq.
(3.111) come to the following asymptotic form (see, for example, Ref.
200):
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
Io(X)
eX
II(X)
(2.rrx)l/2
(when x
>> 1)
53
(3.112)
Thus, if we are interested in the asymptotic behavior of the dependence
of(n)ontimewith(n)>> 1, wehave
n
(3.113)
2
as it should be observed at the ordinary diffusion process. The case is
somewhat different when c < 1. Then the integral entering Eq. (3.110)
tends with --> to a finite value. The integral of Zo(t) taken over the
whole infinite semiaxis is reduced to a sum of tabulated integrals and, as
a result, one obtains the following asymptotes for n ):
(n)
(1
+
o)rPt
(3.114)
--et
which holds true provided, of course, that
rP >3>
(3.115)
a)2
(1
Let us estimate now the relative half-width of the distribution (3.105).
To do this one may derive the equation for the mean-square n 2 ), proceeding directly from Eqs. (3.104). This equation has the form
d
d(n
2
d
n2Zn
2(1
eO’P (n) + (1 + a)’P + e’Pzo (3.116)
Using the above-obtained asymptotic expression for n ), we have the
following asymptotes for the relative half-width of distribution:
d=
((nZ),/2
( rPi0/.)
(n)"
-1
+
1/2
-a
(1
1/2
(3.117)
54
V. N. BAGRATASHVILI ET AL.
,
from which it follows that, with the condition (3.115) fulfilled, d
1,
i.e., the distribution is concentrated near its mean. It also follows from
Eq. (3.117) that the relative width of distribution increases as ct approaches
unity. The distributions shown in Figure 3.13 which are calculated with
the use of exact formulas (3.105) illustrate well the found asymptotic
estimates. These distributions are given for various values of ct, but for
the common n
40. It should be noted that, for ct
0, i.e., when
the rate of upward transitions is negligible compared to that of downward
ones, the distribution (3.105) comes to the Poisson distribution
z.
(rPt)"
exp(-rPt)
n!
(3.118)
Let us now discuss to what extent the picture just considered really
reflects the excitation of the vibrational quasicontinuum of molecules. The
state densities entering the general equations are associated with the sectors
of the vibrational spectrum which are one quantum of radiation apart. If,
for estimates, one neglects the energy of the ground state in the semi0 o6
0.02
gO
6O
0
20
0
FIGURE 3.13 The distributions (3.105) for different values of the parameter x with the
40.
mean energy n
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
55
classical expression (2.70) from Ref. 175, then the ratio of state densities
of two adjacent bands with the energies Evib and Evib + hlq is
p(Evib)
p(Evib d- h’)
(3.119)
Taking the radiation frequency 1000 cm-1 one finds that, for example, for
a seven-atom molecule, the ratio (3.119) does not exceed 0.6 even up to
Evib 40 000 crn-1 which is somewhat higher than the typical dissociation
limits. On the other hand, with ct
0.6 and n
40, one finds from
Eq. (3.117)that d- 0.15. Thus, we should really expect the distribution
produced by radiation to have a distinct maximum and be concentrated
near its mean.
3.5.3. Excitation for the case
of varying cross-sections
Another important point is how the real energy dependence of the crosssection of transitions in vibrational quasicontinuum affects the form of
distribution. In Section 2.3.4 from Ref. 175 we have already discussed
the qualitative shape of absorption bands in the vibrational quasicontinuum.
As the energy increases, the maximum of absorption band is shifted by
anharmonicity to the long-wave region, and the band itself is broadened.
These two points cause the dependencies of the cross section on the number
of transition to be different, in general, for different frequencies of radiation
used. To obtain some simple estimates let us first ignore any changes in
the band shape. This assumption, however, does not mean that the transition cross-section remains constant. Indeed, in our previous paper (Ref.
175) we have derived Eq. (2.81) which enables us to calculate the square
of the dipole moment of transitions from a fixed Evib-energy state integrated
over the whole band and this integrated value, of course, grows versus the
energy. It is logical that just this dependence is tested if one neglects
changes in the band shape, and it will be called below the integral.crosssection law, for short.
For our purposes, it will be convenient here to transform Eq. (2.81)
from Ref. 175 to another form. We consider for simplicity the energy of
the lowest band to coincide with that of the ground state. Then one can
express the cross-sections of all subsequent transitions through the crosssection tro. Indeed, for the cross-sections of any two successive transitions,
one finds using Eqs. (2.81) from Ref. 175 the following equations:
56
V. N. BAGRATASHVILI ET AL.
p,,-
ltr,-
(k
trOl
1,,
pn(r,,,,, +
(k
trOl
k
+ 1)p’[(n
k)hl]
o
k
(3.120)
+ 1)p’[(n
k)hl],
0
where the equality of the field frequency 1" and that Vir of the resonant
mode is taken, and p’ denotes the densities of states constructed by the
rest of modes. Substracting the first of Eqs. (3.120) from the second one,
we get
10n- 10"n-
pnO’n,n +
O’01
1,n
k
or
OnO’o -[-
O’n,n +
E
O’n
p’[(n
k)hl)]
pnO’O1,
0
(3.121)
1,n
Hence it is easy to obtain
O’n,n +
O’01-
Pnk
E
(3.122)
Pk,
0
and, on substituting (3.122) into Eq. (3.101), we come to
dzn
dt
tro,P
Zn-1
Pn
+ lk
Z
.,
Zn
Pk
On
0
k
Zn+
Pn
E
k
P
0
Zn
Pn
k
E
[Ok
0
(3.123)
0
It only remains to include explicitly the dependence of the state density
on the number of band. But first let us clear up some common features of
solutions of Eq. (3.123) concerned with the mean energy and the relative
half-width of distribution. For the mean one may derive directly from Eq.
(3.123), multiplying by n and summing, the following absolutely exact
result:
d(n)_d
dt
dt
or
n
’
crolPt
nZn
o’olP,
(3.124)
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
57
Thus, the mean energy increases proportionally to time. In the same manner
one may find the equation for the mean-square n2 which enters the
expression for the relative half-width of distribution. This equation is
d(n2
d
,
dtn=
n2Zn
+
O’oP(2(n)
,
2
Zn
Onk
Ok),
(3.125)
O
and one can see that it already includes the law according to which the
density of states grows from band to band.
In Section 2.3.1 from Ref. 175 we discussed various approximations
for describing the dependence of the density of molecular states on the
vibrational energy. Suggesting that the features of distribution under strong
excitation are of interest, i.e., when the terms with n -> play the dominant
role in the sum in Eq. (3.125), we take for estimates the semiclassical
approximation [see Eq. (2.70) from Ref. 175] where the energy of the
ground state is neglected. As a result, considering only the two main terms
of the asymptotic expansion, we have
Dnk=O
Pk
n
k
Z
ks
0
n
S
2’
(3.126)
and, on substituting this approximate relation into Eq. (3.125) and using
Eq. (3.124), one finds that
(3.127)
hence the relative half-width of distribution is
d
((n2!) )1/2
n
s-1/2
(3.128)
Recall now that this same value had been already obtained when in Section
2.5.1 from Ref. 175 we discussed the properties of the vibrational Boltzmann distribution at high temperature [see Eq. (2.90) from Ref. 175].
Thus we may conclude that the Boltzmann distribution is a natural
58
V. N. BAGRATASHVILI ET AL.
reference one to be compared with the real distribution produced by radiation in the vibrational quasicontinuum. In addition, we want to point
out one specific case when Eq. (3.123) can be solved analytically, and the
solution really coincides with the Boltzmann distribution. This is the case
when the state densities are approximated by Eq. (2.68) from Ref. 175,
i.e., when a molecule is treated as a degenerate s-dimensional harmonic
oscillator. We do not write out the equations and their solution for this
case to give the reader a chance to practice on his own and convince
himself of the results. We should recall, however, that the representation
of the general Eqs. (3.102) in the form (3.123) implies that the crosssection of successive transitions in quasicontinuum evolves in exact accordance with the integral-cross-section law. In the general case, the change
in the band shape as the vibrational energy grows in what has been discussed
in Ref. 175 (Section 2.3.4) leads, of course, to a different cross-section
law. Here the situation is probably the most realistic when the radiation
frequency lies near the center of the band at the lower transitions. In this
case, because of the anharmonic shift of the band to the long-wave spectral
region, the growth of the real cross-section is slower than that of the integral
one (in particular, the cross-section may even decrease), and one can
understand that the mean energy must grow slower than linearly and the
distribution itself must be narrower than the Boltzmann one. Otherwise,
if at the lower transitions the radiation frequency lies at the long-wave tail
of the band, then the real cross-section grows more rapidly up to a certain
energy than the integral one, and the distribution is perhaps wider than the
Boltzmann one.
3.5.4.
Influence of dissociative decay and rotational structure
Below, in this series of papers, we shall discuss specific numerical calculations of the dynamics of excitation of molecules in the vibrational
quasicontinuum which will be compared with the actual experiments. As
a rule, we shall find support for the rough qualitative picture presented
here. More detailed calculations must involve two more points which are
not hard to include into the calculation procedure.
First, as far as the states above the dissociation limit are concerned, the
terms which describe the decay of a molecule into fragments must enter
the equations. Therefore, a more general form of kinetic equations for the
quasicontinuum dynamics is
59
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
tr._
dt
1,.P zn-
Pn
O’n,n +
[n +
z.
+
1]/
(3.129)
k.z.,
_
k. 0 for the states below the dissociation limit and the decay
k. for the states above the dissociation limit can be calculated with
where
rates
P"
1P(zn
\
the use of the RRKM theory discussed in Section 2.4 from Ref. 175.
Second, so far we have not allowed for rotations. The shape of the
absorption contour in the quasicontinuum, with transitions in various rotational branches involved, has been already discussed in Ref. 175 (Section
2.3.4). For example, for spherical tops as well as for parallel bands of
symmetrical tops, the transitions are possible in three branches from any
vibration-rotation level with the rotational quantum number J. With the
cross-sections of these transitions being designated as cr.’,s. ; 1, o’,s. / l,
and
], one should write the general equations for populations which
allow for the change in J in the following form:
o.;s.
l;s"P z._
at
1,s-
p.
Zn,J
+ On’J-
p
1,n
Z
1,J
(3.130)
:.
Pn
Pn+
P
z.,
Dn+
z.
+
._
+
Dn+
Zn
+
1,J
+
As before, the populations are averaged over the resonant vicinities of
bands and now the notation z.,s refers not only to the n-th band but also
to the rotational quantum number J.
3.5.5. Multiphoton transitions into quasicontinuum
So far, we have treated various fragments of a complex picture of the
molecular spectrum exhibiting elementary one-photon and multiphoton
transitions between the lower quantum levels as well as successive ones
60
V. N. BAGRATASHVILI ET AL.
FIGURE 3.14 The diagram of levels showing a possibility for direct multiphoton transitions
into the quasicontinuum.
within the quasicontinuous spectrum of high vibrational states. We have
repeatedly laid emphasis on the intricacy for insight into the problem of
the behavior of a molecule in the radiation field on the whole. In this short
section we wish to add one more simple fragment. Let us conventionally
divide the energy spectrum of a molecule into two regions (see Figure
3.14): the lower region consisting of several discrete levels, the transitions
between which are quasiresonant with respect to the field frequency, and
the upper one where the transition spectrum is quasicontinuous. In the case
when the Rabi frequencies in the lower-level subsystem exceed the characteristic value of detunings from resonances, the upper level of the lower
subsystem (designated as la )) is excited effectively enough to ensure further
excitation to the quasicontinuum. Let, however, the Rabi frequencies be
much smaller than the characteristic detuning, including the multiphoton
ones. Then the population of the level la) is much less than unity, i.e.,
1. This value can be calculated within perturbation theory and if,
Paa
for example, the lower subsystem consists just of two levels, then
"
61
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
/a
(,] O)aO)2.
Paa
(3.131)
If the lower subsystem consists of three levels, then
)aa
(-
010)2(2"
0)10
OJal)2,
(3.132)
etc. But, despite the inequality [3aa < 1, the excitation efficiency may be
considerable if many irreversible decays of the level la into the quasicontinuum have time to occur during the radiation pulse. The molecule as
if leaks from the ground state into the quasicontinuum. 21 The leakage rate
is given by the following rather evident equation:
W
DaaO’aP,
(3.133)
where O" is the cross-section of the transition from the level la to the
quasicontinuum. Physically this effect is rather clear and, to some extent,
is identical to the well-known quantum-mechanical tunneling through a
barrier but its treatment in terms of the direct multiphoton transition into
the quasicontinuous spectrum is perhaps more essentially accurate.
In Section 3.3 we considered the dynamics of multiphoton transitions
between the discrete states. Here, on the contrary, the upper state is a band
of levels. If they are spaced closely enough then, like the one-photon
process (see Section 3.7), the multiphoton decay is also described by a
simple kinetic equation but in this case one must use the multiphoton Rabi
frequency instead of the one-photon Rabi frequency. Thus, the rate of
multiphoton transition into the quasicontinuum must be proportional to In,
where n is the number of radiation quanta required for the transition. Using
Eq. (3.53) for the multiphoton Rabi frequency, one can derive the following
expression for this rate:
(3.134)
62
V. N. BAGRATASHVILI ET AL.
If there are several channels one should, of course, sum over them in the
manner discussed in Section 3.3.
Some specific estimates will be given in Section 5 of our series of papers,
when discussing various theoretical approaches to the description of excitation of the lower molecular vibrational levels by the laser field. Here
we want just to note that among all treated resonant processes, the multiphoton transition into the quasicontinuum seems to be the only rather
universal one, since it does not require accidental coincidence of the field
frequency with that of any transition. But it is quite obvious from the
estimates on multiphoton Rabi frequencies obtained in Section 3.4 that
this process is probably effective in that case only when the limit of
vibrational quasicontinuum is low in energy.
3.6. Concluding remarks
In this chapter we have treated some questions on interaction of multilevel
molecular systems with radiation. We have tried to embrace the most
principal points such as the resonant criteria for one-photon and multiphoton processes, various approximations for describing the action of radiation on molecules, and the role of some relaxation processes. As will
be seen from our further discussions, some of the experimental observations
do not need anything more explained, but some important results cannot
be interpreted within the simple schemes. More complex theoretical treatments are called for. We shall examine them, but do not wonder if you
do not find in this series of papers a comprehensive explanation for all the
experiments. And all this despite the work of many experts in the theory!
To make the theory more complex is not always the best and most universal
way for insight into the problem. By using the estimates obtained in this
section and correlating them with experiment you will be able yourself to
detect the theoretical points that are open to criticism. Experiment is of
course the chief judge, and we believe that some of our readers will be
able to find key experimental ideas, the theoretical treatment of which will
allow us in the future to replace the modest "elements of theory" in the
title of this chapter by exhaustive "theory".
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
63
SECTION 4. PHYSICS OF INTERACTION OF A POLYATOMIC
MOLECULE WITH AN INTENSE IR FIELD
This Section gives a short phenomenological description of the basic effects
arising from multiphoton excitation of a polyatomic molecule. These effects
are described in more detail in Sections 5 to 7. The qualitative description
of MP excitation and dissociation of polyatomic molecules presented here
is of particular use for a reader who is interested in the applications of this
effect described in the last Sections, 8 and 9. One can turn to these Sections,
omitting the comprehensive discussion of all the details of the effect in
Sections 5 to 7.
4.1. Main stages of MP excitation of polyatomic molecules by
IR radiation
The introduction (Section 1.2) of the first article of this series 22 presented
the simplest model of multiphoton excitation and dissociation of polyatomic
molecules in an intense IR field. According to this model (Figure 1.3 in
Ref. 202) a polyatomic molecule, as the store of its vibrational energy Evib
increases, passes in succession through different energy regions characterized by different densities and width(s) of vibrational-rotational levels:
discrete
vibrational
levels
with Evib
(region I)
quasicontinuum
of vibrational
levels
real
continuum
of vibrational levels
< Eqc with Eqc < Evib < D with Evib > D
(region III)
where Eqc is the lower limit of the vibrational quasicontinuum, or in other
words, the stochastization limit of vibrational energy, and D is the lower
dissociation limit. In Section 2.3 (article I, Ref. 202) consideration was
given to the meaning of a quasicontinuum of vibrational energy and its
lower limit Eqc for polyatomic molecules.
The interaction of a polyatomic molecule with an IR field essentially
depends on which of the vibrational energy regions previously mentioned
it is located in. We are going to discuss it briefly on the basis of the data
given in Section 2 (article I, Ref. 202) and Section 3 of the present article.
64
4.1.1. Role
levels
V. N. BAGRATASHVILI ET AL.
of IR radiation intensity in excitation of lower vibrational
In the region of lower discrete vibrational-rotational levels (region I) a
molecule acted upon by an IR field undergoes successive stimulated resonant transitions on the ladder of vibrational levels. Of particular importance here is that each vibrational level splits into a great number of
rotational and vibrational-rotational sublevels due to a different type of
vibration-rotation interaction (see Section 2.1.5 in Ref. 202). On the other
hand, because of this the molecules are distributed initially over a lot of
sublevels even if they are concentrated in the ground vibrational state.
This, of course, impedes simultaneous excitation of the molecules from
all the sublevels under a monochromatic IR field. On the other hand, such
splitting opens up many potential ways for multistep and multiphoton
excitation of molecules in rather intense IR fields despite of anharmonic
frequency shifts in successive vibrational upward transitions (see Section
2.1 in Ref. 202).
The IR radiation intensity essential for a molecule to pass through the
entire region of lower vibrational levels up to the region of vibrational
quasicontinuum depends substantially on the type of molecule. Indeed, the
type of molecule governs the value of anharmonicity, the character of
splitting of each vibrational level and the position of the vibrational quasicontinuum limit. For a three-atom molecule, for instance, the anharmonic
shift is large, the number of split sublevels for each vibrational level is
small and the vibrational quasicontinuum limit, one may say, is almost
lifted to the dissociation limit (Eqc
D). In this case extremely intense
IR fields (see the calculations in Section 3.2) are required for multiphoton
excitation of such molecules. Therefore, despite many attempts made in
special experiments 203 205 MP excitation and dissociation of three-atom
and especially diatomic molecules has not been observed experimentally
yet.
In another limiting case, on the contrary, for complex polyatomic molecules, the anharmonic shift is moderate, the number of split sublevels is
very large and the vibrational quasicontinuum limit drops very low, down
to the first vibrational level (Eqc hf). In this case it is possible to excite
the molecules from all the ground states directly to the vibrational quasicontinuum by means of one-quantum transitions. Such excitation calls for IR
radiation of very low intensity. The situation is typical, for example, in
the case of isolated molecular ions with the number of atoms no less than
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
65
9, i.e., with the number of vibrational modes S > 20, which fall within
the vibrational quasicontinuum as the radiation intensity is of the order of
1 W/cm2. 206
Most of the polyatomic molecules for which MP excitation is observed
and studied lie intermediately between the above extremes. Unfortunately,
it is impossible to give a universal description for the excitation of molecules in the region of lower vibrational levels (region I) since such excitation depends on the structure of vibrational-rotational levels of the
particular molecule. Therefore, the excitation of molecules in region I has
been so far based most likely on experiment rather than on calculations.
This is due, of course, to the scant spectroscopic data on the molecules
in lower vibrational states except the ground and first excited states. Despite
this, it is possible to formulate certain general considerations on the MP
excitation of polyatomic molecules at lower vibrational transitions.
For a considerable fraction of the molecules distributed initially over
lower sublevels to be involved in effective MP excitation in region I and
to reach the vibrational quasicontinuum (region II), it is necessary that the
IR laser radiation should comply with at least two conditions: 1) the radiation frequency f must lie within the band of vibrational absorption; 2)
the radiation intensity P must exceed some value of P(I)
P > P(I).
(4.1)
The value of P(I) depends greatly on the type of molecule, the accuracy
of tuning the IR field frequency f to the MP excitation resonance and the
fraction of molecules which must be excited to the vibrational quasicontinuum limit. So, understandably, the value of the P(I) parameter varies
over a very wide range. In the case of simple light polyatomic molecules,
for example, in region I, 5 or 6 IR photons should be absorbed if the f
frequency is detuned considerably relative to the exact resonances of suc6. In this case, as
v
0
cessive stimulated transitions v
the calculations in the foregoing section show (Section 3.2.4), the value
of P(I) may be as high as 107 to 109 W/cm2. On the contrary, for heavy
complex molecules P(I) is reduced to 103 to 103 W/cm2. From such wide
variation the P(I) value is rather conventional. It is introduced here only
to emphasize a critical parameter of MP excitation of molecules in the
region of lower vibrational transitions irrespective of their subsequent MP
excitation in the vibrational quasicontinuum.
66
V. N. BAGRATASHVILI ET AL.
4.1.2. Role of IR radiation energy fluence in excitation
quasicontinuum
of vibrational
The molecules which have reached the vibrational quasicontinuum limit
are able to go on absorbing the IR radiation. It is much simpler to describe
the IR MP excitation in region II than in region I because it is a successive
multistep absorption of IR photons. In this region there is no need for an
intense field to compensate for appreciable detunings between the IR field
frequency 12 and the vibrational transition frequency. Therefore, the absorption in region II is universal by nature and can be described in terms
of the absorption cross-section tr(12, Evib) depending on the vibrational
energy Evib (see Section 3.5).
In essence, the absorption of IR radiation by excited molecules in region
II is reduced to vibrational heating of the whole of the molecule, as was
pointed out by Bloembergen. 27 For strong vibrational heating up to energies Evib D it is apparently necessary that absorption saturation at the
transitions of vibrational quasicontinuum should be obtained. In the case
of an isolated molecule which does not lose its energy, this means that
the IR radiation fluence at the 12 frequency must satisfy the condition
(I)
(I)sa (II)
hf
r(f)’
(4.2)
where (I)sa (II) is the energy fluence of a laser pulse saturating the transition
absorption in the vibrational quasicontinuum.
In fact, in the case of intense MP excitation of molecules, when the
number of absorbed IR photons (n) D/h12 >> 1, a more rigid condition
should be fulfilled. It follows from relations (3.118) or (3.124) given in
the previous Section
>
(n) hfl
tr(f)
D
r(f)
(II),
(4.3)
where (II) is the laser pulse energy fluence essential for strong MP
excitation of a molecule to the dissociation limit.
Thus, the excitation of a polyatomic molecule in region II depends not
on the radiation intensity but only on energy fluence. The estimation presented is, of course, qualitative by nature since it does not allow for
variations in the cross-section tr(12, Evib) as molecules are being excited
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
67
(see Section 2.3.4 in Ref. 202 and below in Section 4.3). But it determines
qualitatively correctly the critical parameter (II) of MP excitation of
molecules in the region of the vibrational quasicontinuum.
Since the MP excitation of molecules in regions I and II is governed by
different parameters of IR radiation, intensity P and energy fluence
conditions (4.1) and (4.3) must be fulfilled at the same time to provide
MP excitation of the molecule. This point is considered below in more
,
detail.
Let a polyatomic molecule be exposed to IR radiation for the time "rp
that does not exceed the time of its isolation "riso or its collisions with other
molecules, the walls and so on, that is
Tp
(4.4)
Tisol
Under condition (4.4) only collisionless photoexcitation of the molecule
due to absorption of a large number of IR photons is possible.
For most of the experiments in low-pressure gases (below Torr) "risol
< 10-6 s (see Section 2.5 in Ref. 202), and CO2 laser pulses with their
duration ’l-p
10-7 S are used for irradiation. In these cases the energy
fluence essential for effective excitation of molecules to the vibrational
quasicontinuum > (II) dictates that the pulse intensity be
P
t(II)/,rp
P(I)
(4.5)
And the typical values of (II) in that region lead to radiation intensities
from 106 to 108 W/cm2. At such intensity, condition (4.1) can be fulfilled
easily, i.e., the MP excitation of molecules to region II is always possible.
In other words, in experiments of this kind, the absorption in the vibrational
quasicontinuum is the limiting process of MP excitation.
This manifests itself more vividly in experiments with shorter pulses "rp
10-9 S when, according to (4.5), higher intensities can be attained (108
to 101 W/cm2). Such intensities ensure full excitation of molecules from
all initial vibrational-rotational states to the region of vibrational quasicontinuum. Such experiments were performed in Ref. 208 especially to
ensure MP excitation of the maximum (100%) fraction of molecules.
Figure 4.1 shows the regions of laser radiation intensities and energy
fluences which are essential for MP excitation of molecules according to
conditions (4.1) and (4.3). In a simple way it also shows the shape of the
vibrational energy distribution of a polyatomic molecule in different regions
68
0
V. N. BAGRATASHVILI ET AL.
P(Z)
FIGURE 4.1 Intensities P and energy fluences of IR laser radiation essential for MP
excitation of molecules to the dissociation limit. MP excited molecules lie in the shaded area.
The vibrational distribution of molecules F(Evib) formed from MP absorption is shown in a
simplified way for each of the regions of the parameters (, P).
of the parameters (, P). It may be seen that in the region > (II) the
increase of the P intensity attained due to a reduction in pulse duration
results in more efficient excitation of molecules to region II.
On the contrary, in experiments with long times a’isol, for example, in
the above-mentioned experiments 26 with molecular ions (’l’isol Is), one
can use very long pulses and even continuous radiation without violating
condition (4.4). In this case the energy fluence critical for MP excitation
of molecules in region II can be apparently realized at comparatively low
intensities of IR radiation. The situation realized here is opposite to (4.5)
(II)/’rp
" P(I)
(4.6)
The limiting process here is the excitation of molecules at lower levels
since the low intensity of IR field does not provide MP excitation of
molecules in region II and, therefore, it becomes impossible to observe
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
69
MP excitation. Exceptions to this are molecules with large numbers of
degrees of freedom s for which the lower limit of the vibrational quasi1. In this case there is
continuum probably goes down to the level v
no need for MP excitation of molecules in region I because right after
resonant absorption of one IR photon the molecule enters region II. It is
in the case of such polyatomic molecular ions that strong MP excitation
and dissociation in a continuous IR field of low intensity has been observed. 26
The region of MP excitation and dissociation of molecules (shaded)
shown in Figure 4.1 is highly dependent of the type of a polyatomic
molecule as has been already noted. In the case of comparatively simple
molecules (BCI3, C2H4, etc.) MP excitation necessitates both high intensities and high energy fluences since the values of P(I) and (II) for simple
molecules are rather high (see the data from the next article III). From
practical considerations such laser radiation can be easily produced in the
form of short powerful pulses. And conversely, in the case of MP excitation
of molecules with large numbers of atoms (S2F]o, SFsNF2, etc.) it is
possible to use IR radiation of comparatively moderate intensity and lower
energy fluence since the values of P(I) and (II) for such molecules are
small (see the data in the next article III, in Sections 5 and 6). Such
radiation parameters have been readily available, of course, for a long
time, but their applications have been limited because it has been difficult
to implement condition (4.4) of collisional excitation for a long time. In
experiments with molecular gases at a pressure of about Torr short pulses
have to be used when condition (4.1) is exceeded.
4.2. MP excitation of molecules at lower vibrational transitions
In a sufficiently intense IR field a polyatomic molecule absorbs a considerable number of photons, passing in succession all the considered regions
of vibrational energy. Such MP absorption by a multilevel molecular system features two specific behavior characteristics: l) an even continuous
increase of the number of absorbed IR photons with increasing radiation
intensity; 2) a resonant dependence of MP absorption at varying radiation
frequency. An important parameter is the fraction of molecules participating in MP absorption of IR radiation and caused to enter the region of
the vibrational quasicontinuum. Let us consider briefly these most important characteristics of MP absorption. They will be discussed in more detail
in Section 5 of the next paper III.
70
V. N. BAGRATASHVILI ET AL.
10
1.,0_"
1# ""
/’1,0"
1,0"
C, J,/’cm"
10’
10
10
10’
p, W/cm
-
FIGURE 4.2 Dependences of the average number of IR photons absorbed per one molecule
in the volume under irradiation for SF6 (from Ref. 203), OsO4 (from Ref. 210) and OCS
(from Ref. 203). The values of the IR radiation frequency are given in terms of cm
-.
4.2.1. "Nonsaturable" behavior
polyatomic molecules
of MP absorption of IR radiation by
This property of MP absorption of IR radiation was observed in the very
first experiments. 23’29 Figure 4.2 shows the experimental dependence of
the average number of absorbed IR photons per SF6 molecule in the volume
or radiation intensity P (the
under irradiation* on the energy fluence
pulse duration was constant, ’rp 90 ns). As may be seen, this dependence
*Attention must be drawn to two different notations ((n) and ) for the average number
of absorbed IR photons which are fundamentally different. The (n) value introduced in
Section 3 denotes the average level of vibrational excitation in the ensemble of excited
molecules. The value introduced here denotes the average number of IR photons absorbed
per molecule in the volume under irradiation. These values may not coincide, first, if not
all the molecules participate in MP absorption (in this case (n) > ) and, secondly, if the
initial excitation level is (no) high enough, i.e., when (no) >> 1.
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
71
is continuous, a little slower than a linear dependence. This points to the
1 or simultaexcitation of higher vibrational levels after the level v
neously with it, i.e., to a gradual increase of the average level of vibrational
excitation with increasing intensity P or energy fluence
MP absorption
saturation sets in at >> 1 because of dissociation of the excited molecules
when they fall within region III. Similar dependences were obtained later
for many other polyatomic molecules. They will be considered in more
detail in Section 5 of the next paper III.
The even () dependences observed in the experiments seem at first
to disagree with the sequence of two different stages of MP excitation
presented in Section 4.1. Indeed, let the molecule be in any initial state.
With small intensities (or small ) the dependence of must increase
linearly until the transition v 0 v
is saturated. Then because of
the absence of the exact resonance of the IR field frequency f with the
successive vibrational transitions v
E/hf
0-- v
v
MP absorption must be low up to the intensities P P(I) which compensate
for the frequency detuning of successive stimulated transitions. In other
words, the (P) dependence in the region P < P(I) must be quite nonlinear
as it is schematically shown in Figure 4.3. With P > P(I) the energy
growth of a molecule due to the transitions in the vibrational quasicontin-
.
III
D
FIGURE 4.3 Simplified dependence of the excitation level on laser pulse energy fluence
for molecules in different initial states having a different degree of resonance with the IR
field in the region of lower vibrational transitions.
72
V. N. BAGRATASHVILI ET AL.
uum must be linear by character since, according to the calculations in
Section 3.5, d(n)/d
trqc(fll).
Strictly speaking, the () dependence can actually be nonlinear due
to the fact that the absorption cross-section trqc in region II depends on
vibrational energy. More accurately the growth of vibrational energy of a
molecule can be found from the equation
d(n)
d
rq (,(n)).
(4.7)
To determine its out-of-linearity in region II it is sufficient to know the
qc (-,Evib) dependence. Continuous increase of the number of absorbed
photons (n) must take place up to (n) D/hl when the dissociation of
molecules in region III becomes essential.
As may be seen from Figure 4.2, no well-pronounced bend can be
observed, in fact, in the experimental dependence for SF6. In other words,
there is no material difference between the MP absorption at the lower
vibrational transitions and in the vibrational quasicontinuum. This can be
explained by the initial distribution of molecules over many rotational levels
in the state v
1. Indeed, a small fraction of molecules being in an exact
multistep resonance with the IR field has a low value for P(I) and quickly
falls within region II. Further vibrational excitation in region II, however,
is possible just at sufficiently large values of > (I)sa (II). For any other
fraction of molecules the multistep resonance with the IR field in region
I is not so exact and, therefore, considerable intensity is required for their
MP excitation. Yet the MP excitation of all the molecules in region II
comes about almost in the same way by (4.7). The observed smooth
dependence () can result from summing up all such groups of molecules
having a different degree of frequency detuning in region I (in Figure 4.3
they are shown by the dashed line in region I).
This explanation is supported by some experiments in which a deviation
from the smooth dependence shown for the SF6 molecule in Figure 4.2
was observed. The () dependence for SF6 at low temperature, for example, is faster than the linear one. TM This is accounted for by a reduced
number of initial quantum states of the molecule under cooling and hence
a decreased number of potential pathways of successive vibrational-rotational transitions of the molecule. In this case two- and three-photon pro-
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
73
cesses of excitation directly to the vibrational state v
2,3 begin to be
involved, their probability nonlinearly dependent of intensity. Another
example: the accurately measured () dependence for a molecule with
comparatively few atoms, such as OsO4 is stepwise by character with
several ranges of saturation. -1’212 This dependence is illustrated in Figure
4.2. It is explained by the fact that different groups of molecules in different
initial states penetrate region II at quite different field intensities. This
gives rise to steps in the dependence.
At the same time, the () dependence for simple molecules (OCS,
D20, etc.) measured in Ref. 203 is similar by character to the saturated
absorption of a two-level system with a maximum number of absorbed
photons < (Figure 4.2). This means that for such molecules the value
of P(I) critical for MP absorption at lower vibrational transitions is too
high and cannot be attained in experiments. In essence, the () dependence measured for a molecule (see Section 5 of the next paper III) gives
a good and reliable idea of the response of the molecule to MP dissociation
without the need for direct observation of dissociation.
4.2.2. Resonant character
of MP absorption
The dependence of the number of absorbed photons on the frequency
called the MP absorption spectrum has a well-pronounced resonant behavior. It is just this property that enables selective MP excitation of
molecules and particularly its application for laser isotope separation.
Therefore, the resonant characteristics of MP absorption are important in
practice and their research receives primary attention. By now measurements have been taken of the MP absorption spectra for dozens of molecules, mainly in the region of 9 to 11 txm of a CO2 laser. The basic
features of MP absorption spectra are: high sensitivity to radiation intensity,
broadening and "red" resonance shift with increasing intensity, and formation of distinct narrow resonances in the spectrum in a certain intensity
range.
Figure 4.4 shows the evolution of the MP dissociation spectrum for the
SF6 and OsO4 molecules as the intensity increases. In the case of SF6 one
can observe a well-pronounced shift of the absorption maximum to the
long-wave length side and a broadening of the absorption band. This result
was obtained in Ref. 203 by the opticoacoustic measurement of MP absorption and later was confirmed in Ref. 213 by measuring the transmission
74
V. N. BAGRATASHVILI ET AL.
7"-
/
2.0
d
1.5
’
//
/
0.5
o
935
./o
940
,.,dE
x
//
/
RIO
[
/
<n>
PIO
P30
"o
,/
/
P-
/
’
/"
945
(cm -I)
=
q
/
/
I
950
955
920
90
960
9
[cm’l
FIGURE 4.4 Evolution of the MP absorption spectra of SF6 and OsO4 molecules (300 K)
with increasing IR pulse energy or intensity. Solid curves show the linear absorption
spectrum: a) curves 2, 3, 4, 5 and 6 are attained at radiation intensities of 0.035, 0.15, 1.2,
10 and 60 MW/cm with 100 ns. The scale of the curves on the ordinate must be increased
by the factor given for each curve (from Ref. 203); b) curves 2, 3, 4 and 5 correspond to
intensities of 3.5, 7.1, 10.5 and 14 MW/cm (from Ref. 210).
of a gas cell. The shift and broadening of the MP absorption band with
increasing intensity can be qualitatively accounted for by vibration anharmonicity. As the IR radiation intensity increases, the average level of
molecular excitation and hence the contribution of higher vibrational transitions to MP absorption increase, too. Since the spectrum of transitions
between high vibrational levels due to anharmonicity shifts to the longwave length side with an increase in the number, this brings about a gradual
"red" shift and broadening of the MP absorption band.
It must be noted that at room temperature only 30% of the molecules
are in the ground vibrational state while the rest are under some equilibrium
distribution over the other low-lying vibrational levels. This gives rise to
"hot" bands in the linear absorption spectrum (the long-wave length peaks
on the solid curve in Figure 4.4a). The distribution of molecules over many
initial states greatly complicates, of course, the MP absorption spectrum
since many different pathways of MP excitation of the molecule become
possible. When SF6 is cooled down, this leads to a narrowing of the linear
IR absorption spectrum24 because of the decreased intensity of"hot" bands
and the decreased width of molecular distribution over rotational levels.
75
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
This changes at once the MP absorption spectrum in which some structure
appears. 211,215
In some cases the structure in the MP absorption spectrum appears in a
definite intensity range: at room temperature, for example, in the case of
the OsO4 molecule (Figure 4.4b). 21 With the IR pulse intensity P > 30
kW/cm2 for OsO4 the absorption varies from the linear one (Figure
4.3) and three peaks appear in the MP absorption spectrum at the CO2
laser lines. Their position agrees with the successive anharmonic shifts
5.98 cm-1) of the v3 mode under excitation. With an increase
(2 [X33
of P the position of the peaks does not change but their contrast is reduced
and the MP absorption band width increases.
The MP absorption spectra given in Figure 4.4a,b have been taken at
comparatively low resolution (about 2 cm-) in experiments with discrete
tuning of the CO2 laser oscillation frequency at different rotational-vibrational lines. It is of greater interest to produce MP absorption spectra with
higher resolution. Such experiments were performed in Ref. 216 with the
CO2 laser with its continuous tuning of generation
frequency. In experiments with C2H4 molecules the formation and evolution of the narrow resonances of MP absorption were studied. With the
laser radiation energy fluence ranging from 10-3 to 10-2 J/cm2 (or the
intensity varying between 104 and 105 W/cm2), one could observe the
formation of very narrow resonances of MP absorption just 10-2 cm wide.
The appearance of such resonances shows that the MP excitation of molecules at lower vibrational transitions does occur very effectively because
of the multistep and multiphoton resonances in a multilevel system. Narrow
resonances of MP absorption have also been obtained for the SF6 molecule
with the use of a continuously tunable CO2 laser. 27
In conclusion it should be emphasized that the resonant absorption of a
large number of IR photons in these cases is a property of any isolated
polyatomic molecule even in the absence of any collisions. This is an
essential difference from the resonant absorption of IR radiation by a
molecular gas when there are many collisions between the molecules during
the irradiation time. In that case the absorption of a large number of IR
photons per molecule has a trivial explanation: vibrational energy exchange
between molecules during collisions (see Figure 1.2b). Regardless, MP
excitation of a polyatomic molecule can be observed distinctly under condition (4.4), i.e., in the absence of any collisions, for example, in experuse of a high-pressure
-
iments with molecular beams 28’219 and molecular ions isolated in vacuum
with the use of electric and magnetic fields. 26
76
4.2.3. Fraction
V. N. BAGRATASHVILI ET AL.
of MP excited molecules
In the previous section when discussing the collisionless absorption of a
large number of IR photons by a polyatomic molecule we had to deal with
the concept of a relative fraction of molecules participating in MP absorption. This concept is a basic one in interpreting MP excitation and
dissociation of molecules and must be considered specially. At small IR
pulse intensities when the MP excitation of vibrational levels is not yet
dominant, the monochromatic IR field in the absence of rotational relaxation interacts only with a small fraction of molecules at the rotational J
at the field
0- v
levels for which the resonant transition v
frequency f/is possible. The thermal population of these rotational levels
determines the maximum fraction of molecules f that can be excited by a
short pulse to the level v
as the vibrational-rotational transition is
saturated. This effect called the "rotational bottle neck" was predicted in
Ref. 192 and observed experimentally in Ref. 193. In the foregoing Section
(Section 3.3.2) it is shown how this effect has influence on the excitation
of the vibrational-rotational transition. The value of f for different molecules lies in the range between 0.1 and 0.001 and is highly dependent on
gas temperature.
In the first experiments on MP absorption29’22 attempts were made to
allow for this effect for estimating the real number of photons absorbed
by the SF6 molecules involved in the process of MP excitation. For this
purpose the absorbed energy was distributed only among the f-th fraction
of molecules in the region of irradiation. Such a rough estimate caused
the number of IR photons absorbed by a molecule to be initially overstated
greatly. In Ref. 221, by observing the bleaching of the IR absorption band
simultaneously at many vibrational-rotational transitions it was concluded
that many initial rotational states were depleted under the action of a
powerful IR pulse. This experiment shows that the fraction of SF6 mole10-2 to
cules q() involved in the process of MP absorption with
2
1.0 J/cm is much higher than the fraction of molecules f interacting with
the IR field in the linear limit, i.e.,
q(CI)) >>. f
q(O)
(4.8)
The most important question, however, is to what extent the value of
q(f/) is close to unity at some intensity or energy fluence. The answer to
this question was given in experiments with OSO4, 222 SF6223 and CF3I. 224
77
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
20" Hot" molecules
16
,,
-I
0
o)
"Cold" molecules
n(E)
0.05
b)
0.1
0,2
0.5 1,0
( J/cm 2
2.0
FIGURE 4.5 Formation of two ensembles of vibrational "hot" and "cold" molecules under
MP excitation by powerful radiation pulses: a) distribution of vibrational energy of molecules
at MP excitation; b) dependence of the relative fraction of highly excited molecules q on the
energy fluence of CO2 laser pulse for SF6 (from Ref. 223), OsO4 (from Ref. 222) and CF3I
(from Ref. 224).
In these experiments different methods were applied to find that the value
of q() at MP excitation grew with an increase of from comparatively
small magnitudes to its maximum (q
1). Thus, it has been shown that
under typical conditions MP excitation forms two ensembles of molecules.
Some part of molecules q involved in the process of MP excitation forms
an ensemble of vibrationally "hot", i.e., highly excited, molecules. The
rest, part (1
q) of molecules, populates the lower vibrational levels and
forms an ensemble of vibrationally "cold" molecules which probably do
not differ materially from their initial state (Figure 4.5a).
In other words, it has been found that the function of the molecular
vibrational energy distribution F(Evib) after MP excitation can be approximately presented as
F(Evib)
q()Fhot(Evib) q- (1
q)Fo, (Evib),
(4.9)
where Fcold(gvib) is the vibrational distribution describing the ensemble of
"cold" molecules, and Fhot(Evib) is the vibrational distribution of highly
excited molecules describing the ensemble of "hot" molecules. The average
number of absorbed IR photons introduced above for all the molecules
78
V. N. BAGRATASHVILI ET AL.
n h’
Evib [F(Evib)
F0(Evib)] dEvib
is related to the average number of photons
molecules of the "hot" ensemble
nq h[
Evib [Fhot(Evib)
q
(4.10)
(n) absorbed by the
Fcold(Evib)] dEvib
(4.11)
by the simple relation
((I),[)
q ((I),") q(,l)),
(4.12)
where it is assumed that the average energy of the molecules in the hot
ensemble q h" is much higher than the average energy of the molecules
in the cold ensemble and Fcod (Evib) Fo (Evib).
Figure 4.5b shows the dependences of the fraction of molecules q excited
to the vibrational quasicontinuum on the energy fluence for three molecules obtained in Refs. 222-224. It can be clearly seen that with the IR
pulse energy fluence ranging from 0.1 to 1.0 J/cm 2, typical of most experiments, only a fraction of molecules falls within the vibrational quasicontinuum. Therefore, it is impossible to measure correctly the average
energy of the excited molecules qh’ Eq without taking into account
the effect of "rotational" bottle neck or the effect of formation of two
ensembles of molecules in our case. But as the energy fluence in the case
of the above-mentioned molecules increases to several J/cm2, it can easily
be seen that all the molecules are involved to MP excitation, i.e., the only
1, there is
"hot" ensemble is formed. Under these conditions, when q
no need to take into account the effect of formation of two ensembles. To
overcome the indeterminacy in estimating the fraction of molecules q
involved in MP excitation and to measure correctly the absorbed energy
Bloembergen et al. 28 used a subnanosecond pulse of CO2 laser for MP
J/cm2 such a pulse
excitation. Even at a moderate energy fluence
9
has high intensity (over 10 W/cm2), and therefore MP excitation of all
the molecules, that is, the realization of the condition q
1, could be
expected. This allowed measuring the dependence of the dissociation yield
79
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
of molecules on the energy from the IR field without measuring the value
of q.
Now that we know that the average number of absorbed IR photons per
one molecule in the volume under irradiation fi observed in the experiment
is, in fact, the product of the fraction of excited molecules q by the average
level of their excitation fiq, it is of interest to follow the origin of the
resonant dependence (). Such measurements of q(f) and q(f) have
been carded out for two molecules: OsO4222 and CF3I. 225 Figure 4.6 shows
the dependence q(l), obtained in Ref. 222 with a fixed value of IR pulse
1.2 J/cm2. At the maximum of linear IR absorption
energy fluence (I)
0.6 and drops drastically at the long-wave length edge of the
band q
band. Figure 4.6 also shows the frequency dependence of the average
energy q of the molecules in the "hot" ensemble. The number of IR
photons absorbed by the "hot" molecules, on the contrary, increases at the
long-wave length edge. This is due to an increase of the absorption crosssection in the vibrational quasicontinuum within the long-wave region.
p t,O
30
2O
10
"0.2
_-01
Z
0.5-
-0.04
-002
0
920
’
I"
91.0
960
980
[crn"1
FIGURE 4.6 Dependence of the relative fraction q of highly excited OsO4 molecules and
their average energy Eq
"qhl) (in terms of dissociation energy) on the l frequency of IR
1.2 J/cm and gas pressure of 0.03 Torr (from
radiation pulse with its energy fluence (I)
Ref. 222).
80
V. N. BAGRATASHVILI ET AL.
Thus it can be concluded that the resonant character of MP absorption
spectra, i.e., the (12) value determined, according to (4.12), from the
product of () by q(12) is related to the resonant dependence of the fraction
of molecules penetrating to the region of vibrational quasicontinuum.
4.3. MP excitation of molecules in the vibrational
quasicontinuum
The MP excitation of polyatomic molecules in region II, that is the vibrational quasicontinuum, materially depends on the strong interaction of
a large number of vibrational modes. Such interaction, first, gives rise to
a vibrational quasicontinuum in which, instead of narrow resonances of
MP absorption, a wide smooth absorption resonance appears around every
vibrational mode which is active in IR absorption. Second, mixing of
vibrational modes results in the stochastization of vibrational energy absorption among all vibrational modes. This specific feature of MP excitation
of the vibrational quasicontinuum is qualitatively considered below.
4.3.1. Evolution
quasicontinuum
of MP absorption spectrum in the vibrational
The dependence of the MP absorption spectrum on the IR field frequency
can be clearly seen from the experimental results for the SF6 and OsO4
molecules in Figure 4.4. The experimental dependences (11) observed,
however, depend on the MP absorption spectrum both at lower vibrational
transitions and in the vibrational quasicontinuum. It is very important to
obtain separately the data on the MP absorption spectrum in regions I and
II.
The first direct experimental data on the vibrational quasicontinuum of
polyatomic molecules and the transition spectrum in it was obtained from
the experiments 226’227 on MP dissociation of molecules in a two-frequency
IR field. A comparatively weak IR field with its frequency fl resonantly
excited the molecules from the ground vibrational state into the vibrational
quasicontinuum, and the second rather intense IR field, with its frequency
f12 detuned from the resonance frequency of unexcited molecules, performed MP excitation of molecules in the vibrational quasicontinuum up
to the dissociation limit. In this method the functions of resonant excitation
of molecules at lower transitions and excitation in the vibrational quasicontinuum between two pulses of different frequencies and intensities are
separated. This technique allows the separate study of the transition spec-
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
eslnd
ple)(J ]A
81
82
V. N. BAGRATASHVILI ET AL.
trum in the vibrational quasicontinuum. The frequency dependence of the
dissociation yield 13 can give information on the excitation efficiency of
the vibrational quasicontinuum at different frequencies.
Figure 4.7a shows the general idea of MP excitation and dissociation
of a polyatomic molecule in a two-frequency IR field and the results of
the first experiment226,27 for the SF6 molecule. In Figure 4.7b one can
clearly see the resonance dependence of MP excitation on the ’1 frequency
of the first laser pulse interacting with the molecules at lower vibrational
transitions. At the same time the dependence of the dissociation yield on
the ’2 frequency of the second pulse (Figure 4.7c) is less obvious. In this
experiment the CO 2 laser frequency 2 was tuned unfortunately only to
the blue region about the absorption band of SF6. Therefore, it may possibly
be the nonresonant MP excitation of SF6 molecules due to absorption at
the short-wave length wing of the wide absorption band in the vibrational
quasicontinuum that was observed. In subsequent experiments with SiF4, 228
SF6228’229 and OsO4230 one could observe a large shift of the maximum of
the 13(fl2) dependence to the long wave length spectral region.
It should be noted that, owing to the exponential dependence of the 3
dissociation yield (see Section 2.4 in Ref. 202) on absorbed energy, a
comparatively small increase of the absorption cross-section in the vibrational quasicontinuum brings about a large increase of 13. Therefore, for
direct measurement of the absorption spectrum in the vibrational quasicontinuum it is better to measure the frequency dependence of the energy
fluence (II) which causes strong excitation of the transitions in the vibrational quasicontinuum and hence molecular dissociation. Such measurements were taken for OsO4230 and SF6, 228 and the value of (II) was
found to decrease from about J/cm2 to 0.1 J/cm2 as 2 was red-shifted.
The experiments on MP dissociation of molecules in a two-frequency
IR field as well as the experimentTM on measuring the IR absorption
spectrum of heated SF6 gas (Figure 2.12 in Ref. 202) show that the spectrum
of excited molecule in the vibrational quasicontinuum is concentrated near
its frequencies active in IR absorption. The maxima of these bands shift
to the long-wavelength side because of vibration anharmonicity as the Evb
vibrational energy increases.
In some Wol’ks 225’232 the models of absorption spectrum in the vibrational
quasicontinuum are considered (see Section 2.3.4 in Ref. 202). It has been
found that the experimental data can be described with the use of the
Lorentzian contour for the spectral density of the square of the matrix
element I,l2(Evib) or absorption cross-section tr(O, Evib)
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
O’qc(’’Evib)
O’int(Evib)"ff
[[-
J(Evib)
Vmax(Evib)] 2
+
2
(Evib)
83
(4.13)
where Vmax(Evib) and (Evib) describe the center frequency and the halfwidth of the absorption band from a state with energy Evib in the vibrational
quasicontinuum. The simple model representations given in Section 2.3.4202
permit approximate estimations of the energy dependences O’in (Evib), (Evib)
and Vmax(Evib).
First, the integral absorption cross-section O’int(Evib) gradually increases
as the vibrational energy is increased. This is explained, of course, by an
increase in the vibrational amplitude of a highly excited molecule. Figure
2.13 in Ref. 202 presents the calculated dependences 12(Evib) or O’in (Evib)
on the vibrational energy Evib for the molecules CF3I and SF6. With Evi
increased by 20 000 crn-1, the O’in cross-section increases by about 2 or 3
times.
Second, the absorption maximum frequency Vmax (Evib) gradually shifts
to the long-wavelength region with increasing vibrational energy. The value
of such a shift, according to relation (2.82) from Ref. 202 depends linearly
on the value of vibrational energy, and the coefficient of such a shift is
determined by the anharmonicity constants.
Third, the absorption band width (Evi) grows with an increase of
vibrational energy since the anharmonic interaction for highly excited states
increases. The law of increase of (Evi) is determined by Fermi resonances,
i.e., by the interaction of molecular vibrations. The dependence (Evib)
should be treated specifically for every given molecule. For the CF3I
molecule, for example, it is apparently the three-frequency Fermi resonances that play the basic part. Therefore, in Ref. 225 the dependence
(Evib) 3/2
vib is taken for this molecule.
To sum the data presented, Figure 4.8 shows in a simple way the twodimensional dependence ([l, Evib) which gives a qualitative indication of
the potential for MP excitation of a polyatomic molecule. The experimental
dependence of the number of absorbed IR photons in a one-frequency field
(ll) results, roughly speaking, from successive superposition of such
absorption spectra as the vibrational energy Evib increases. Therefore, the
dependence (ll) substantially shifts to the red region (Figure 4.4) with
increasing radiation intensity which corresponds to an increase in Evib. In
much the same way the decreasing of growth of the number of absorbed
photons with increasing energy fluence is related to the red shift of
absorption band. As a result, the absorption cross-section for highly excited
84
V. N. BAGRATASHVILI ET AL.
FIGURE 4.8 Qualitative evolution of the IR absorption spectrum of a polyatomic molecule
in the vibrational quasicontinuum near the vibrational band as the vibrational energy of the
molecule increases.
.
molecules at the fixed frequency f drops and their further excitation reSo the experimental dependence ()
quires the higher energy fluence
(see Figure 4.2) is usually somewhat weaker than the linear dependence
that might be expected in the absence of an absorption band shift (Figure
4.3).
From the qualitative dependence o’,c(f, Evit,) it can easily be seen that
the efficiency of MP excitation and dissociation of a polyatomic molecule
in a two-frequency IR field at properly chosen frequencies f and f2 is
much higher. The excitation pathway of the molecule in this case is shown
by arrows in Figure 4.8. Thus, the IR absorption spectrum of an excited
molecule r(I, Evib) is a key characteristic for interpreting the MP excitation
of a molecule. The quantitative description of molecule MP excitation calls
for direct experimental data on the absorption spectrum of molecules in
the vibrational quasicontinuum, at least the dependences of 1)max(Evit,) and
(Evit,). For this purpose, new methods of laser spectroscopy of highly
excited vibrational states should be developed. In this case one can apparently apply the described method of excitation of molecules by two IR
and f2.
pulses with different frequencies
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
4.3.2. Distribution
85
of vibrational energy of a molecule under MP
excitation
A powerful IR radiation pulse usually acts on a molecule distributed over
many initial states, i.e., those under different conditions. Under the action
of an IR pulse the distribution of vibrational energy in the molecular
ensemble becomes nonequilibrium. That is why two questions may arise:
1) what is the intermolecular distribution of vibrational energy in the
ensemble of molecules under irradiation; 2) what is the intramolecular
distribution of vibrational energy both over energy levels and over vibrational modes?
The intermolecular distribution of vibrational energy formed under a
powerful IR pulse is essentially nonequilibrium due to the formation of
two molecular ensembles" vibrationally excited molecules in the region of
the vibrational quasicontinuum (a hot ensemble) and those left at the lower
vibrational levels (a cold ensemble) as is shown in Figure 4.5.
In two cases, when the vibrational temperatures differ greatly, it is
possible to avoid the formation of two such ensembles. First, when the
molecules are excited by a very powerful short pulse with intensity P >>
P(I) it is possible to excite all the molecules to the vibrational quasicontinuum, i.e., to realize the condition q 1.208 Second, a buffer gas inducing
strong rotational relaxation of the excited molecules during a laser pulse
but not causing their appreciable vibrational deactivation can be added into
the molecular gas. This can always be done since the probability of change
of rotational energy, as a polyatomic molecule comes into collision with
an atom of the buffer gas, is much higher than the probability of change
of vibrational energy. In this case collisions make it possible to suppress
the effect of rotational bottle neck during MP excitation of molecules from
many rotational states. Finally, after the laser pulse is over, the collisions
of "vibrational-hot" and "vibrational-cold" molecules result in vibrational
energy exchange between two ensembles and their combining into one
ensemble with the same total vibrational energy.
The most direct information on intermolecular distribution of vibrational
energy has been obtained in the experiments 233 by the measurement of the
Raman scattering spectrum in the vibrations of molecules excited by a CO2
laser pulse. Figure 4.9 presents the results of these experiments which
vividly confirm the basic physical effects in MP excitation of molecules.
At the top there is a Raman scattering spectrum in the Stokes region at
the vl vibration of SF6 molecules without any excitation. When the molecules are excited by a CO2 laser pulse, an additional peak belonging to
86
V. N. BAGRATASHVILI ET AL.
v
o)
c=
b)
c=05 J / cm
0
co- 0.5
J/cm ?
d) %d =Sons
120
Pxe
,,
tort.
700
750
800
VL-VR[Cm1]
3/-,70
"-’ 360
357@
FIGURE 4.9 Spectra of Raman scattering Stokes signal of the second harmonic of ruby
laser at the v vibration of the SF6 molecule: a) unexcited SF6 molecules; b-d) the molecules
are excited by a CO2 laser pulse with its energy fluence of 0.5 J/cm2. The delay time of the
ruby laser probe pulse about the CO2 laser exciting pulse a’a 50 ns (b, d) and a’a 4 txs
(c). The last spectrum (d) is produced in the presence of buffer gas: Xe, 120 Torr (from Ref.
233).
the ensemble of hot molecules appears in the Raman scattering spectra
(Figure 4.9b). This peak is a little smaller due to the anharmonic vibration
shift and is shifted towards the Stokes region. This spectrum unambiguously points to the formation of two ensembles of vibration-excited molecules fight after the CO2 laser pulse action is over. The ensembles combine
into one because of vibrational exchange which is evident from the peak’s
disappearance when the delay between the CO2 laser pulse and the probing
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
87
pulse for exciting the Raman scattering spectrum is increased to 3 Ixs
(Figure 4.9c). With a buffer gas added, rotational relaxation during MP
excitation provides penetration of all the molecules from the lower levels
into the vibrational quasicontinuum and only one ensemble of vibrationally
hot molecules is formed (Figure 4.9d).
Intramolecular distribution of vibrational energy in the ensemble of hot
molecules excited to region II is formed under the action of two processes:
1) stimulated upward and downward transitions under laser radiation; 2)
mixing of vibrational energy between modes due to their interaction. In
the presence of collisions, of course, particularly after the laser pulse is
over, a Boltzmann distribution is expected to be attained in the hot molecular ensemble.
The shape of the vibrational distribution of molecules in the hot ensemble
has not been so far measured in experiments. On the basis of the dissociation yield one can state indirectly that it really differs from the Boltzmann
one. This agrees well with the calculation that, in the region of vibrational
quasicontinuum, is rather reliable. The vibrational energy distribution in
the region of the vibrational quasicontinuum can be calculated on the basis
of the kinetic equations presented in the previous Section (Section 3.5).
An essential feature of the MP excitation of a polyatomic molecule in
region II is the increasing density of vibrational states as Evib grows.
According to the simple estimate given in Section 3.5.2., the ratio of state
densities, with the vibrational energy changing by the hfl value which
corresponds to absorption of one IR photon, increases by (1 + h/Evib) s-1
times, where s is the number of vibrational degrees of freedom. Therefore,
the stimulated upward transitions for a polyatomic molecule (Evib
Evib
+ hfl) prevail considerably over the downward transitions (Evib + h
Evib). Both the analytical 232’234 and numerical 196’218’219’225’235’236 computations show that for the real values of probability ratio of upward and
downward transitions the distribution form F(Evib) is rather close to the
Poisson distribution (3.118). The difference in level density is only by
factors of 1.5 to 2 times as the molecular energy changes by the energy
of one CO2 laser quantum hi) turns out to be rather essential.
Vibrational distribution calculated in Ref. 225 using equation (3.123)
and Boltzmann distribution at equal values of absorbed energy and at two
values of laser pulse energy fluence. It may be seen that the calculated
distribution of vibrational energy is somewhat "narrower" than the Boltzmann one and has an exponential "tail" that falls more rapidly. It can be
easily understood from the note at the end of Section 3.5.3. Indeed, it
88
V. N. BAGRATASHVILI ET AL.
should be noted that the absorption cross-section in the vibrational quasicontinuum tr(, Evib) for CF3I has a red anharmonic shift Vma (Evit).
Therefore, when the IR radiation frequency is tuned to the center of the
band at the lower transitions, the value of the cross-section tr(12,Evib) for
highly excited transitions at the same frequency I gradually falls (see
Figure 4.8). This leads to a "compression" of the vibrational distribution.
The difference of this vibrational distribution tail from the tail of the
Boltzmann distribution at the same average number of absorbed photons
n is quite observable in the experiment. The point is that it is the tail
of the vibrational distribution lying above the dissociation limit D that
contributes to the dissociation yield of the molecule (see Section 4.4). So
the difference of the real distribution from the Boltzmann one manifests
itself well in the dependence of the dissociation yield 13 on absorbed energy.
After the laser pulse action is over, the Boltzmann distribution of populations in the hot ensemble of molecules must arise due to collisions.
This gives a gradual Boltzmann tail in the distribution and can make a real
contribution to the molecular dissociation after the laser pulse.
Vibrational energy stochastization of highly excited molecules in the
region of the vibrational quasicontinuum has already been discussed in
Section 2 (Section 2.3) of the foregoing paper. 22 This property of highly
excited molecules follows from very general principles of molecular dynamics. The vibrational motion is stochastic when the vibrational excitation
energy Evib reaches the dissociation energy D. This conclusion indirectly
follows from numerous measurements of the dependence of the molecular
dissociation yield on the energy absorbed by a molecule. The results of
such measurements (see Section 4.4 and the more comprehensive Section
7 in paper IV of the present series) are in good agreement with the statistical
theory of monomolecular decay that is based materially on the assumption
of vibrational excitation stochasticity.
The most important point, however, which is still not clearly understood
is the position of the stochasticity energy limit Eqc. This problem is of
importance for molecular spectroscopy, multiphoton laser chemistry and
the theory of nonlinear vibrations of systems with many degrees of freedom.
Section 2 (2.7) of the previous article 22 is concerned with potential
spectroscopic methods of measurement of the vibrational energy distribution. It is shown that the integral absorption factor in a vibrational band
that is active in the IR spectrum must be weakly sensitive to the type of
vibrational distribution and to the energy in this vibrational mode. Such a
weak dependence is related only to the anharmonic terms responsible for
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
89
the difference in the probability of upward and downward vibrational
transitions. From this standpoint one should approach carefully the interpretation of a number of experiments 237-239 based on the method of double
IR-IR resonance. The authors believe that they observed the effect of
vibrational energy stochastization in a molecule. In these experiments the
SF6 gas was excited by a powerful laser IR pulse and the IR absorption
of the probe laser beam was measured with a time resolution.
There are two processes for the reliable study of the vibrational motion
stochastization in molecules where the molecular upward and downward
transitions should be discriminated: spontaneous IR emission and antiStokes Raman scattering (RS). As is shown in Section 2 (Section 2.7) of
the previous paper, 2 the integral intensity of the spontaneous emission
radiation band in the mode active in the IR spectrum and the integral
intensity of the anti-Stokes Raman scattering band in the mode active in
RS are proportional to the vibrational energy in the mode.
It is the method of spontaneous RS that has been applied for the direct
observation of the vibrational energy stochastization in a highly excited
molecule and to estimate the stochastization limit of vibrational energy.
In this experiment, with the average energy of the SF6 and CF3I molecules
in the hot ensemble varying in the vicinity of 10 000 cm-1, the anti-Stokes
Raman scattering from the second harmonic of ruby laser has been observed
at all vibrational modes active in the Raman scattering spectrum. This
unambiguously points to the excitation of these modes, and the excitation
level remained constant as the probe pulse delay about the exciting pulse
was varied in a range from 20 ns to 10-6 S. At the same time, with a long
delay a’a the vibrational energy distribution over all the modes must definitely have reached equilibrium from collisional vibrational exchange.
Thus, the vibrational motion stochastization under the conditions of
experiment4 occurs for a time shorter than 20 ns, i.e., during the very
process of MP excitation. On the basis of such measurements 24 it has
been concluded that the energy limit of vibrational motion stochastization
in the SF6 molecule is 3 300 cm-3 higher than the initial vibrational energy
at 300 K, Eo
4. For a simpler
700 cm i.e., in the region of v3
6 000 cm-1. The
molecule, like CF3I, such measurements give Eqc
methods of anti-Stokes RS seem to be used in systematic measurements
of Eqc for many polyatomic molecules.
The excitation of molecules to high vibrational states lying in the region
of the vibrational quasicontinuum by IR radiation is closely related to the
problem of the response of nonlinear systems with many degrees of freedom
-,
90
V. N. BAGRATASHVILI ET AL.
to an external periodical force. This problem has been the goal of studies
for a long time (see Ref. 242). The possibility of systematic studies into
polyatomic molecules is of fundamental interest and these smallest nonlinear vibrational systems with many degrees of freedom, different in their
structure and accessible in unlimited amounts for experimentalists, particularly in studying the energy limit and the time of full stochastization of
vibrational energy, are very important.
The time of vibrational energy stochastization "rstoch depends on anharmonic interaction and lies probably in the picosecond range. This range
of IR pulse duration is very difficult to attain as yet for experimental
measurements of’rstoch. Nevertheless, it has several promising possibilities.
Specifically, of great interest is the situation when not all the vibrational
degrees of freedom but only a certain part of them participates in the
formation of transitions in the vibrational quasicontinuum near the frequency of the mode being excited. Such a situation, for example, may
take place in complex molecules he,ving spatially separated different functional groups. The anharmonic interaction between the vibrational modes
of these functional groups may turn out to be rather weak in this case.
The situation can exist when there is not an appropriate Fermi resonance
for some molecular mode that provides its coupling with the rest of the
modes. In both cases it is probably possible to observe incomplete stochastization of vibrational motion in a molecule even for a time of about
10-9 S. The probability of such incomplete stochastization, of course,
increases considerably in the subnanosecond range of pulse duration.
4.4. Dissociation of MP excitation of molecules in continuum
A highly excited molecule within region III, that is, the region of a real
continuum, may be chemically transformed even without collisions with
other reacting partners due to various monomolecular reactions: dissociation, isomerization, dissociative ionization, etc. In this section we are going
to restrict ourselves to considering the basic features of the most important
and perfectly understood monomolecular process, that is, the dissociation
of highly excited molecules.
4.4.1. "Threshold" character
of dissociation yield
The dissociation yield, i.e., the probability of dissociation of a molecule
acted upon by one IR radiation pulse has two very characteristic features:
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
91
a sharp, almost threshold, dependence on the laser pulse energy fluence
and a resonance dependence on the laser pulse frequency f. The dependence 13(,f) is restricted, of course, by the corresponding dependences
of the MP absorption rate. These characteristics are considered in more
detail below.
Even in the very first work, 243’244 where the effect of visible luminescence of molecules in a strong IR field in the absence of collisions was
revealed, the steep threshold character of the dependence of the luminescence intensity on the laser pulse intensity was observed. The threshold
properties of the effect of MP dissociation were first studied comprehensively in Ref. 245 with the SF6 molecule. The dependence of the dissociation yield 13() on the laser pulse energy fluence obtained in this work
is given in Figure 4.10a. It may be seen that with
2.0 J/cm2 one can
observe a distinct threshold of molecular dissociation. The decrease of the
o,
a
o,6
(l
,o
,o
,o 5,o
,..7/C e
FIGURE 4.10 Dependences of the MP dissociation yield 13 for the SF6 molecule (a) and
the MP dissociation efficiency q for the CF3I molecules (b) on IR laser pulse energy fluence:
a) 13 in arbitrary units (from Ref. 247); b) the solid line stands for experimental and the
dashed line for calculated (from Ref. 225).
92
V. N. BAGRATASHVILI ET AL.
pulse energy by only 8% did not bring about any appreciable dissociation
even though, with the energy fluence being below the threshold, the number
of pulses essentially increased. With the threshold exceeded, the dependence of the dissociation yield was 3 3. In Ref. 246 the dissociation
yield was measured as the SF6 molecule was acted upon by CO2 laser
pulses of 0.5 ns and 100 ns duration. It was shown that the dissociation
threshold was determined by the energy fluence rather than the intensity
of the laser pulse. It must be said that the work changed the terminology.
Before, the MP dissociation threshold had been related to laser radiation
intensity (see, for example, Refs. 243-245 and 247. Figure 4.10a from
Ref. 245 is given with the more correct terms.
In the first works dealing with the theory of MP dissociation some
attempts were made to explain the threshold dependence. In these works
the threshold property of the effect was associated with the molecule
passing the lower levels lying below the vibrational quasicontinuum limit
due to power broadening 27’248 or with multiphoton transitions 249’25 which
are believed to reflect the exponential function 13(). But under such
explanations the same steep dependence on should have been observed
for the value of absorbed energy () as well which does not correspond
at all to the experimental data (see Section 4.1).
The key to the correct explanation of the threshold property of the effect
of MP dissociation was found in Ref. 247 where the dependences 13()
were compared for the SF6 molecule as the bands of the v3 mode active
in IR absorption and the weak composite vibration v2 + v6 were excited.
It was found that, even though the value of dissociation yield in the case
of v2 + v6 was much smaller than for v3, the thresholds for the two cases
were almost identical. This experimental fact allows the threshold property
to be associated with the excitation of molecules to the vibrational quasicontinuum which does not depend on how easily a molecule passes through
the energy range below the vibrational quasicontinuum limit. In other
words, under standard experimental conditions the energy fluence (II)
that ensures effective excitation of molecules from region II to region III
can be realized at intensities P
P(I) which are quite sufficient for a
significant fraction of molecules to pass from region to region II (see
Section 4.1). Excited molecules have vibrational energy distribution F(Evib)
with a sharp exponential fall-off towards high energies and an average
number of Evib proportional to the energy fluence
At increasing
energy fluence a small fraction of molecules in the vibrational distribution
tail falls at first within the region of the real continuurn. Then it exponen-
.
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
93
tially grows as (I) increases. When the dissociation yield is measured in a
limited region and with limited sensitivity, such an exponential dependence
of the dissociation yield looks like a threshold dependence. More precise
measurements give, of course, just the exponential dependence of the
dissociation yield 13((1)) or 13(( n )) (Figure 4.10a). This explanation of MP
dissociation yield is now considered correct for molecules whose vibrational quasicontinuum limit lies not so high and which, therefore, dissociate
at quite accessible laser pulse energy fluence (1 to 100 J/cm).
If the laser energy fluence exceeds the threshold, the dissociation yield
increases rapidly. When the value of 13 becomes considerable, for example,
about 0.2-0.3, the growth of 13((1)) slows down and then tends to saturate.
Figure 4.10b illustrates such a dependence for the CF3I molecule. It should
q)
CFa
,/
1- Experiment
2- Theory
0
0
b
-
X o 3.8 t 0.2 cm-1 Vo= 0.73 -+ 0.03 cm -1
III|
0
(I) J/cm 2
FIGURE 4.10 (continecD
94
V. N. BAGRATASHVILI ET AL.
be noted that in this figure, instead of the dissociation yield per pulse 13,
on the ordinates the value of the so-called quantum efficiency of MP
dissociation is plotted
q
D
13 hl)
(4.14)
Physically it reflects more correctly the process of the dissociation of a
highly excited molecule and is suited for comparison of theory with experiment. The point is that the dissociation yield per pulse is affected by
two factors" the fraction of molecules q falling within region II and the
dissociation probability of a highly excited molecule depending on the
absorbed energy q -/q (see Section 4.2.3). The value of q is apparently
independent of q since both a q and g are proportional to the fraction of
molecules q falling within the vibrational quasicontinuum.
Since the absorption cross-section in the vibrational quasicontinuum,
according to (4.13), depends on the IR radiation frequency, it follows from
the approximated expression (4.3) that the energy fluence providing the
excitation of molecules in region II to the limit of region III is frequencydependent, too. Therefore, to be precise, the "threshold" energy fluence
providing a significant dissociation yield should also depend on the
frequency. The conclusion of Ref. 247 on the same dissociation threshold
of SF6 for the bands v3 and v2 + v6 has been specified in Ref. 246 showing
that in the case of the band v2 + v6 the threshold is about twice as high
than in the case of the band v3. Moreover, even within one absorption
band the change of the absorption cross-section in region II causes the
threshold to change for different laser frequencies. 451’452
The dependence of the MP dissociation yield 13 on the MP absorption
efficiency of IR radiation can be clearly seen in similar spectral dependences
of dissociation yield and absorbed energy. Figure 4.11 shows the dependence of the dissociation yield of 32SF6 on the pulse laser frequency 1) as
it is excited within the v3 band. 245 The maximum of this dependence, as
well as that of its corresponding dependence of absorbed energy (see Figure
4.4a) is shifted to the long-wavelength spectral region. Even though the
spectra of MP dissociation 13(f) and MP absorption B(I)) are much wider
than the IR absorption linear spectrum, in many cases the width of
is considerably smaller than the isotope shift of absorption bands of different isotopic molecules. A typical example of such a situation is presented
in Figure 4.11 for two isotopes (32S and 34S) of the SF6 molecule. 245 From
95
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
O.O1o.
0.02-
/"r"
0.03>"
I
O.O&910
920
930
[cm
I
SF
-
9/0
S
950
FIGURE 4.
Dependence of the MP dissociation yield of SF on the IR radiatio pulse
frequency explaini te effect of isotopic selectivity of MP dissociation: 1) experimental
dependence p() in bitra units for the SF molecule, 2) assumed dependence p() for
3SFo; 3) and 4) spectra of line absotion for 32SF6 and SF respectively (from Ref. 24).
the dependences presented it may be seen that as SF6 containing a mixture
of such isotopic molecules as 32SF6 and 34SF6 is irradiated, there is no
dissociation selectivity at the radiation frequency fo at which the relative
dissociation yields of both isotopic molecules are equal. As the frequency
is shifted to the long-wavelength spectral range about o, preferential
dissociation of 34SF6 must take place, and as it is shifted to the shortwavelength region, preferential dissociation for 32SF6 molecules occurs.
The p(f) dependences play a key role in choosing a radiation frequency
as the effect of IR MP dissociation is applied as a method of laser isotope
separation.
4.4.2. Overexcitation
of polyatomic molecules in continuum
Under the action of a powerful IR pulse the nonequilibrium distribution
of molecules over vibrational levels sets in (see Section 4.3.2). The distribution tail reaches the limit of a real continuum of states which corresponds to molecular dissociation. If the vibrational energy of a molecule
exceeds the dissociation energy D or in other words, the energy of breaking
of the weakest bond do, the molecule is able to dissociate into fragments.
The dissociation rate must evidently depend on the molecular state. For
example, if the vibrational excitation energy is localized at the bond do,
the dissociation must occur very quickly, during about one period of vibration. But if a considerable portion of energy is in other vibrational
degrees of freedom, dissociation calls for fluctuation as a result of which
96
V. N. BAGRATASHVILI ET AL.
the energy would be accumulated at the bond do. It takes a longer time
than one period of vibration to realize this critical fluctuation. This is
described by the expression (2.85) for monomolecular dissociation rate
given in Section 2.4 of the previous paper. 22
The simple formulas in Section 2.4 of Ref. 202 make it possible to
estimate how strong the molecule can be excited over the dissociation limit
due to its interaction with a laser IR pulse under typical experimental
100 ns. If the energy fluence of
conditions when the pulse duration "rp
a laser pulse exceeds the threshold value
(II) not very strongly, the
number of absorbed IR photons is equal to the ratio D/h12. In this case
the rate of passing of each successive transition connected with the absorption of one IR photon may be estimated as ko (D/hl)’r-1. Since
the typical values of D/h12 range from 20 to 50, ko (2-5) 108 s-1. Due
to a rapid increase of the dissociation rate k (Evib) the characteristic values
of the Evib energy to which a molecule can be excited by an IR pulse can
be estimated from the relation ko k(Evib) that corresponds approximately
to f(Evib)
10-4. Figure 2.14 in Ref. 202 shows the dependence f(Evib)
for different number s of the vibrational degrees of freedom. It may be
seen that for five-atom molecules the values f(Evib) 10-4 can be obtained
even with (Evib
D) < 0.10, for six-atom molecules--with (Evib D)
< 0.20 and for seven-atom molecules--with (Evib D) < 0.30. Thus,
for molecules with the number of atoms N < 7 the excitation over the
dissociation limit under the action of a pulse with a-p 10-7 s can actually
be equal to (0.1-0.3) D. For more complex molecules and/or shorter pulses
the value of overexcitation may even exceed the dissociation energy D.
This is an important feature of IR photodissociation under transient pulsed
conditions.
According to the statistical theory of dissociation, the dissociation products observed generally correspond to breakings of the weakest bond in a
molecule. Direct diagnostics of the radicals resulting from MP dissociation
of molecules became possible with the introduction of the molecular
beam. 218’253’254 This was used in Ref. 218, for example, to prove that in
the case of SF6 first an atom of F breaks away and then the dissociation
of the SF5 radical gives rise to the SF4 radical. The alternative dissociation
mechanism SF6 SF4 + F2 was rejected since there were no F2 molecules
detected. Measuring the function of the kinetic energy distribution of the
resulting radicals g (Evib) enabled the authors of Ref. 218 to obtain information on the vibrational energy of the resulting radicals and hence on the
degree of excitation of dissociating SF6 molecules over the dissociation
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
97
4-10 J/
limit. For example, with the energy fluence of a laser pulse
2
cm the excess of Evib over D comes to 6 to 10 photons of CO2 laser. This
agrees well with the conclusions of statistical theory.
As discussed in Section 2.4 of Ref. 202, two different types of molecular
dissociation are possible with their potential curves shown in Figure 4.12.
In the first case the dissociation is not associated with rearrangement of
chemical bonds, i.e., does not demand that the energy barrier should be
overcome. This usually occurs when one atom breaks away, for example,
an atom of F from SF6 molecule or an atom of I from the CF3I molecule.
In this case the dissociation products have excess vibrational energy (Evib
D). In the other case, on the contrary, the chemical bonds get rearranged
which requires that the energy barrier Eac should be overcome. In this case
a molecule must absorb an additional number of IR photons to overcome
the energy barrier. The excess vibrational energy (Eac D) is transferred
to the dissociation products. Such a case is characteristic of the breaking
from a polyatomic molecule, for example a diatomic molecule, the atoms
of which are not directly bound in the original molecule. A typical example
for this case is the CF2C12 molecule that can dissociate not only through
the channel CF2C12 CF2C1 + C1 (Figure 4.12a) but also CF2C12 CF2
+ C12 (Figure 4.12b). The dissociation probability of this molecule through
the first channel is about one order of magnitude higher than through the
E
R
0
a.
FIGURE 4.12 Schemes of potential curves leading to dissociation of a polyatomic molecule
without a potential barrier (a) and in the presence of a potential barrier with its energy
and with transfer of excess energy pma to the dissociation products (b).
98
V. N. BAGRATASHVILI ET AL.
second one. 255’256 This effect is caused by the presence of a potential barrier
for the second reaction. This was proved by the measurement of the vibrational energy distribution for the CF2 radical in Ref. 257. It has been
found that the vibrational temperature of this radical remains constant (To
1050 K) although the radical yield changes by an order of magnitude
as cI) varies. This points to the fact that molecular dissociation takes place
near the top of the barrier the height of which is 6 900 crn
Finally, we should emphasize an important peculiarity of MP excitation
of polyatomic molecules with a large number of atoms, that is, their
significant excitation over the dissociation limit D. In experiments this
could be clearly observed 258 in measuring the number of IR photons absorbed by one (CF3)3CI molecule. The number of absorbed IR photons
observed for this molecule varies between 35 and 40 which is about twice
as much as the dissociation energy of the weakest bond CmI. At the same
time, similar measurements for such a simpler molecule as CF3I result in
20-22, i.e., the number of IR photons absorbed in the real vibrational
continuum (Evib
D)/hl < 2. 225
This particular feature of polyatomic molecules is essential during MP
excitation by very short pulses when very high vibrational excitation of
molecules over the dissociation limit can occur. In this case the degree of
overexcitation (Evib
D) may be sufficient to overcome the next dissociation limits of a polyatomic molecule, and then the dissociation of the
molecules through many channels becomes possible.
-.
4.5. MP excitation of the electronic states of polyatomic
molecules by intense IR radiation
Vibrationally highly excited molecules in the ground state can pass to
excited electronic states. This can be due to nonadiabatic interaction of
two electron terms (see Section 2.7 in the foregoing article22). The diagram
of electron terms with possible V
E transition is illustrated in Figure
2.19. 2o2 Some fraction of molecules can perform such transitions according
to the difference in densities of vibrational states of molecules with their
vibrational energy Evib in the ground electronic state po(Evib) and molecules
with their vibrational energy (Evib
El) in the excited electronic state
E), where E is the minimum energy limit of excited electron
P(Evib
term. Between the molecules in two electronic states equilibrium sets in
from direct E
V and reverse V E transitions in accordance with the
lifetimes of molecules with their vibrational energy Evib in the ground
-
-
MULTIPLE PHOTON IR LASER PHYSICS. CHEMISTRY. II
99
electronic state "to(0, Evib) and molecules with their vibrational energy (Evib
El) and electronic energy E1 in the excited electronic state "r (El, Evib
E):
po(Evib)
’to(0, Evib)
pl(Evib
"rl(E, Evib
E)
(4.15)
If the excited electronic state is not metastable, usually "r(E1, Evib El)
"to(0, E,,ib). Therefore, even with the vibrational level densities differing
greatly [po(Evib) p(Evib E)] a considerable fraction of vibrationally
excited molecules can pass to an excited electronic state. In experiments
this can be clearly seen as visible luminescence of molecular gas and
appears even in the absence of dissociation. In essence, this process of
conversion of vibrational energy to electronic energy (V E conversion)
is typical of polyatomic molecules. Transitions with the electronic states
of a molecule excited by IR radiation are often called "reverse nonradiative
transitions". Such a process has been observed and studied for a number
of polyatomic molecules, for example, OsO4 .259’260
In the excited electronic state a polyatomic molecule can perform induced
upward transitions under the action of the same IR pulse. This is quite
possible provided that the vibrational band of IR absorption remains in the
excited electronic state. As a result, the molecule can be excited above
the dissociation limit D1 of the excited electronic term. In this case the
dissociation of the molecule will give a molecular fragment in the excited
electronic state (Figure 4.13a). Thus, the luminescence of molecular gas
under powerful pulsed IR irradiation has, in principle, two origins: 1)
luminescence of the original molecule in the excited electronic state; 2)
luminescence of the dissociation products of the molecule through their
excited electronic states. These processes can exist and compete even in
one molecule, as has been observed, for example, in the cases of OsO4259
and CRO2C1226 molecules.
As noted in the foregoing Section 4.4, a polyatomic molecule with a
large number of atoms may undergo strong excitation over the dissociation
limit. In this case mixing of the vibrational states of a molecule in the real
continuum for the ground electron term with the excited electron term may
take place. Specifically, even the V
E transition of a highly excited
molecule to the repulsive electron term of the molecule, that leads to an
almost instantaneous dissociation of the molecule, is possible (Figure 4.13b).
Due to a very short lifetime of the molecule "rl at the repulsive electronic
100
V. N. BAGRATASHVILI ET AL.
o
Ro
FIGURE 4.13 Different schemes of excitation of electronic states at MP vibrational excitation of a polyatomic molecule. The minimum energy of excited electronic state may lie
below (a) and above (b) the dissociation limit.
term the process E
V almost does not come about, and, in fact, one
can observe the electronic predissociation of the molecule due to the mixing
of the nonparticipating electronic terms. Such a process was experimentally
observed in Ref. 261 for the (CF3)3 CI molecule when electron-excited
atoms were formed due to MP excitation of the original molecule.
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